[1] Simulation Study 1: Significance of Regression

[1.1] Introduction

In this simulation I will investigate the significance of the regression test. I will simulate 2 different model

Below is the 1st model, which is the Signifianct model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \epsilon_i.\]

where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)

We have provided the value of the parameters as

  • β0 = 3
  • β1 = 1
  • β2 = 1
  • β3 = 1

Below is the 2nd model, which is a Non-Signifianct model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \epsilon_i.\]

where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)

We have provided the value of the parameters as

  • β0 = 3
  • β1 = 0
  • β2 = 0

  • β3 = 0

  • σ ∈ (1,5,10), three distinct values of sigma

  • n = 25

I will simulate 2500 simulations for both the models for each value of sigma, so around 2500 X 3 = 7500 for each model, so TOTAL 2 X 7500 = 15,000 times for both the model

I need to show there are around 15000 times the model is being trained as part of the simulations

[1.1.1] Goal

As part of this exercise, the following are the Goals to achieve

  • Simulate the MLR models (2 models) for 2500 times for a given value of σ
  • Plot the empirical distribution of the for the following 3 parameters
    • F statistics
    • p-value
    • R square
  • Discuss on the following points in the Disucssion section
    • Do we know the true distribution of the following ?
      • F statistics
      • p-value
      • R square
    • How do the empirical distributions compare with the true distribution ?
    • How R2 is related to the sigma

In order to achieve these goals, we have various methods in the form of R codes, markdown

[1.2] Methods

Below is the code which will runs the 2500 simulations against each sigma (σ)

  • Will Create 2 matrix one for each model (significance and non-significance) which will store the F statistics, p-value and R2 for each sigma value (1, 5 and 10)
  • Once the 2 matrix are populated, will be used to plot the empirical distribution, as well as to compare with the true distribution

[1.2.1] Simulation function for the both signifiance and non-signifiance model

birthday = 19770411
set.seed(birthday)

#Variable declaration
n = 25
sigma = c(1,5,10)
no_loop = 2500

#function to generate samples for both significance and non-significance model
mlr_sim = function(n_in,sd=1,signifincance_ind=1){
  exer_data = read.csv("study_1.csv")
  
  epsilon = rnorm(n_in,mean = 0,sd = sd)
  if (signifincance_ind){
    beta_0 = 3
    beta_1 = 1
    beta_2 = 1
    beta_3 = 1 
    exer_data[,1] = beta_0 + beta_1 * exer_data[,2] + beta_2 * exer_data[,3] + beta_3 * exer_data[,4] + epsilon
  }
  else {
    beta_0 = 3
    beta_1 = 0
    beta_2 = 0
    beta_3 = 0 
    exer_data[,1] = beta_0 + beta_1 * exer_data[,2] + beta_2 * exer_data[,3] + beta_3 * exer_data[,4] + epsilon
  }
  exer_data
}
[1.2.2] Simulation for 2500 iterations (Significance & Non-Significance Model)
birthday = 19770411
set.seed(birthday)

# Below code is to simulate the Significance Model 
model_significance = cbind(sigma_1_F=rep(0,no_loop),sigma_1_P=rep(0,no_loop),sigma_1_R_SQUARED=rep(0,no_loop),
                           sigma_5_F=rep(0,no_loop),sigma_5_P=rep(0,no_loop),sigma_5_R_SQUARED=rep(0,no_loop),
                           sigma_10_F=rep(0,no_loop),sigma_10_P=rep(0,no_loop),sigma_10_R_SQUARED=rep(0,no_loop))
extra_counter = 0

#Loop against each sigma to generate the anova for each simulation and populate the model_nsignificance
for(s in 1:length(sigma)){
  #print(paste("Sigma:",sigma[s]," extra_counter",extra_counter))
  for (i in 1:no_loop){
     mlr_data = mlr_sim(n,sigma[s],signifincance_ind = 1)
     null_model = lm(y~1,data = mlr_data)
     full_model = lm(y~x1+x2+x3,data = mlr_data)
     model_significance[i,s+extra_counter] = anova(null_model,full_model)$F[2]
     model_significance[i,s+extra_counter+1] = anova(null_model,full_model)$'Pr(>F)'[2]
     model_significance[i,s+extra_counter+2] = summary(full_model)$r.squared
  }
  extra_counter = extra_counter + 2
}
# Below code is to simulate the Non Significance Model 
model_nonsignificance = cbind(sigma_1_F=rep(0,no_loop),sigma_1_P=rep(0,no_loop),sigma_1_R_SQUARED=rep(0,no_loop),
                           sigma_5_F=rep(0,no_loop),sigma_5_P=rep(0,no_loop),sigma_5_R_SQUARED=rep(0,no_loop),
                           sigma_10_F=rep(0,no_loop),sigma_10_P=rep(0,no_loop),sigma_10_R_SQUARED=rep(0,no_loop))
extra_counter = 0
#Loop against each sigma to generate the anova for each simulation and populate the model_nonsignificance
for(s in 1:length(sigma)){
  print(paste("Sigma:",sigma[s]," extra_counter",extra_counter))
  for (i in 1:no_loop){
     mlr_data = mlr_sim(n,sigma[s],signifincance_ind = 0)
     null_model = lm(y~1,data = mlr_data)
     full_model = lm(y~x1+x2+x3,data = mlr_data)
     model_nonsignificance[i,s+extra_counter] = anova(null_model,full_model)$F[2]
     model_nonsignificance[i,s+extra_counter+1] = anova(null_model,full_model)$'Pr(>F)'[2]
     model_nonsignificance[i,s+extra_counter+2] = summary(full_model)$r.squared
  }
  extra_counter = extra_counter + 2
}
## [1] "Sigma: 1  extra_counter 0"
## [1] "Sigma: 5  extra_counter 2"
## [1] "Sigma: 10  extra_counter 4"

[1.3] Results

As part of the results we will draw out the following

  • Simulate the MLR models (2 models) for 2500 times for a given value of σ
  • Plot the empirical distribution of the for the following 3 parameters
    • F statistics
    • p-value
    • R2
[1.3.1] Empirical distibution for 2500 simulations (Sigma = 1)
par(mfrow=c(1, 2))

hist(model_significance[,1],
     main = "Empirical Distribution of F [Significance]",
     cex.main = 0.8,
     xlab = "Simulated values of F",
     col = "darkolivegreen",
     border = "white",
     probability = TRUE
     )

hist(model_nonsignificance[,1],
     main = "Empirical Distribution of F [Non Significance]",
     cex.main = 0.8,     
     xlab = "Simulated values of F",
     col = "plum4",
     border = "white",
     probability = TRUE
     )

par(mfrow=c(1, 2))

hist(model_significance[,2],
     main = "Empirical Distribution of P [Significance]",
     cex.main = 0.8,     
     xlab = "Simulated values of P",
     col = "sienna2",
     border = "white",
     probability = TRUE
     )

hist(model_nonsignificance[,2],
     main = "Empirical Distribution of P [Non Significance]",
     cex.main = 0.8,     
     xlab = "Simulated values of P",
     col = "slateblue1",
     border = "white",
     probability = TRUE
     )

par(mfrow=c(1, 2))

hist(model_significance[,3],
     main = "Empirical Distribution of R Squared [Significance]",
     cex.main = 0.7,     
     xlab = "Simulated values of R Squared",
     col = "yellow",
     border = "darkolivegreen",
     probability = TRUE
     )

hist(model_nonsignificance[,3],
     main = "Empirical Distribution of R Squared [Non Significance]",
     cex.main = 0.7,     
     xlab = "Simulated values of R Squared",
     col = "tan",
     border = "white",
     probability = TRUE
     )

[1.3.2] Empirical distibution for 2500 simulations (Sigma = 5)
par(mfrow=c(1, 2))

hist(model_significance[,4],
     main = "Empirical Distribution of F [Significance Model]",
     cex.main = 0.8,
     xlab = "Simulated values of F",
     col = "darkolivegreen",
     border = "white",
     probability = TRUE
     )

hist(model_nonsignificance[,4],
     main = "Empirical Distribution of F [Non Significance]",
     cex.main = 0.8,
     xlab = "Simulated values of F",
     col = "plum4",
     border = "white",
     probability = TRUE
     )

par(mfrow=c(1, 2))

hist(model_significance[,5],
     main = "Empirical Distribution of P [Significance]",
     cex.main = 0.8,
     xlab = "Simulated values of P",
     col = "sienna2",
     border = "white",
     probability = TRUE
     )

hist(model_nonsignificance[,5],
     main = "Empirical Distribution of P [Non Significance]",
     cex.main = 0.8,
     xlab = "Simulated values of P",
     col = "slateblue1",
     border = "white",
     probability = TRUE
     )

par(mfrow=c(1, 2))

hist(model_significance[,6],
     main = "Empirical Distribution of R Squared [Significance]",
     cex.main = 0.6,
     xlab = "Simulated values of R Squared",
     col = "yellow",
     border = "white",
     probability = TRUE
     )

hist(model_nonsignificance[,6],
     main = "Empirical Distribution of R Squared [Non Significance]",
     cex.main = 0.6,     
     xlab = "Simulated values of R Squared",
     col = "tan",
     border = "white",
     probability = TRUE
     )

[1.3.3] Empirical distibution for 2500 simulations (Sigma = 10)
par(mfrow=c(1, 2))

hist(model_significance[,7],
     main = "Empirical Distribution of F [Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of F",
     col = "darkolivegreen",
     border = "white",
     probability = TRUE     
     )


hist(model_nonsignificance[,7],
     main = "Empirical Distribution of F [Non Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of F",
     col = "darkolivegreen",
     border = "white",
     probability = TRUE     
     )

par(mfrow=c(1, 2))

hist(model_significance[,8],
     main = "Empirical Distribution of P [Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of P",
     col = "darkgreen",
     border = "white",
     probability = TRUE     
     )

hist(model_nonsignificance[,8],
     main = "Empirical Distribution of P [Non Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of P",
     col = "darkgreen",
     border = "white",
     probability = TRUE
     )

par(mfrow=c(1, 2))

hist(model_significance[,9],
     main = "Empirical Distribution of R Squared [Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of R Squared",
     col = "darkcyan",
     border = "white",
     probability = TRUE     
     )

hist(model_nonsignificance[,9],
     main = "Empirical Distribution of R Squared [Non Significance Model]",
     cex.main = 0.6,
     xlab = "Simulated values of R Squared",
     col = "darkcyan",
     border = "white",
     probability = TRUE     
     )

[1.4] Discussion

As part of the introduction, we have the following discussion agenda - Do we know the true distribution of the following ? - F statistics - p-value - R square - How do the empirical distributions compare with the true distribution ? - How R2 is related to the sigma

[1.4.1]Do we know the true distribution
[1.4.1.1]True distribution of the F Test
curve(df(x, df1=3, df2=21),
      xlab = "Probability",
      ylab = "Frequency",
      main = "True Distribution of F Test(F Dstribution)",
      lwd=3,
      col = "darkgreen")

The True distribution of the F Tests shows as a F distribution, which is right skewed distribution

[1.4.1.2] True distribution of the P value
hist(pnorm(rnorm(25,mean = 3, sd = 1),mean=3,sd=1),
     xlab = "Probability",
     ylab = "Desnity",
     main = "True Distribution of P val (Uniform Distribution)",     
     breaks = 10,
     col = "orange",
     border = "white",
     probability = TRUE)

The True distribution of the P seems to have an uniform distribution

[1.5.1.3]True distribution of the R squared
curve(dbeta(x, (4-1)/2, (25-4)/2),
      xlab = "Probability",
      ylab = "Frequency",
      main = "True Distribution of R Squared (Beta distribution)",      
      lwd = 3,
      col = "palevioletred4")

The True distribution of R Squared is a beta distribution (right skewed distribution), where the shape 1 and shape 2 parameter is of value (k-1)/2 and (n-k)/2, where k = number of esimator in the True model(which is 4,i.e \(\beta_{0} , \beta_{1} , \beta_{2} , \beta_{3}\)) and n = number of elements in the data (which is 25)

[1.4.2] How do the empirical distributions from the simulations compare to the true distributions?

[2] Simulation Study 2: Using RMSE for Selection?

[2.1] Introduction

In this simulation project we will investigate the procedure for choosing the best model via test/train RMSE by simulating the following multiple regression model (MLR)

\[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \beta_6 x_{i6} + \epsilon_i.\]

where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)

We have provided the value of the parameters as

  • β0 = 0
  • β1 = 5
  • β2 = -4
  • β3 = 1.6
  • β4 = -1.1
  • β5 = 0.7

  • β6 = 0.3

  • σ ∈ (1,2,4), three distinct values of sigma

  • n = 500

We need to evaluate 9 models (below), where for each model, we will calculate the RMSE for train data set (250 random observation) and test data set (250 random observation). We will simulate this for 1000 iteration for each value of σ, so there would be of 3000 simulation for each model, which equals to 3 X 3 X 3000 = 27000 for around 9 models

  • (Model -1 ) y ~ x1
  • (Model -2 ) y ~ x1 + x2
  • (Model -1 ) y ~ x1
  • (Model -2 ) y ~ x1 + x2
  • (Model -3 ) y ~ x1 + x2 + x3
  • (Model -4 ) y ~ x1 + x2 + x3 + x4
  • (Model -5 ) y ~ x1 + x2 + x3 + x4 + x5
  • (Model -6 ) y ~ x1 + x2 + x3 + x4 + x5 + x6, the correct form of the model
  • (Model -7 ) y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7
  • (Model -8 ) y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8
  • (Model -9 ) y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9

The Train/Test RMSE for each model is as below

RMSE(model, data) = \(\sqrt{\dfrac{1}{n}\sum\limits_{t=1}^{n}(y_{i} - \hat{y}_i)^2}\)

As per the above model, the correct model is the Model -6, we need to see whether by this methods/approach, we will able to get the correct model

[2.1.2] Goal:

As part of this exercise, the following are the Goals to achieve

  • Simulate the MLR models for 1000 times for a given value of σ (1,2,4)
  • Plot 3 curves, each plot is against each value of sigma, shows what the Train RMSE and Test RMSE against the Model Size (Model 1,2,3,4,5,6).
  • From the 3 Plots, figure out whether the simulation helped to identify the best Model, i.e. Model 6

  • Discuss on the following points in the Disucssion section

    • Does the simulation method always select the correct model (i.e. model 6), or sometimes it select the other models
    • If we do the average of the Test RMSE, whether the simulation method selects the correct model (i.e. Model 6)
    • How does the level of noise (σ = 1,2 and 4) affects the Model selection.

In order to achieve these goals, we have various methods in the form of R codes, markdown

[2.2] Methods

Below is the code which will runs the 1000 simulations against each sigma (σ)

  • Will Create 9 matrix for Train Set which will stored the Train RMSE, where each columns holds the Train RMSE for each value of σ = 1,2 & 4
  • Will Create 9 matrix for Test Set which will stored the Test RMSE, where each columns holds the Train RMSE for each value of σ = 1,2 & 4
  • Will Create 9 matrix which stored the average Train Set RMSE from the Train Set RMESE, where each columns holds the Train RMSE for each value of σ = 1,2 & 4
  • Will Create 9 matrix which stored the average Test Set RMSE from the Test Set RMSE, where each columns holds the Train RMSE for each value of σ = 1,2 & 4

[2.2.1] Code for Simulations

birthday = 19770411
set.seed(birthday)

#Declaration
beta_0 = 0
beta_1 = 5
beta_2 = -4
beta_3 = 1.6
beta_4 = -1.1
beta_5 = 0.7
beta_6 = 0.3
n = 500
no_loop = 1000
sigma = c(1,2,4)
sig_data = read.csv("study_2.csv")

#Function to generate the sample data
mlr_sim = function(data,sd=1){
  epsilon = rnorm(n,mean = 0,sd = sd)
  data[,1] = beta_0 + beta_1 * data[,2] + beta_2 * data[,3] + beta_3 * data[,4] + beta_4 * data[,5] + beta_5 * data[,6] + beta_6 * data[,7] + epsilon
  data
}

#Defining the Train Matrix to stored the TRAIN RMSE
train_rmse_model1_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model2_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model3_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model4_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model5_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model6_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model7_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model8_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model9_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))

#Defining the Test Matrix to stored the TESE RMSE
test_rmse_model1_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model2_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model3_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model4_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model5_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model6_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model7_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model8_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model9_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_mat = cbind(sigma_1=rep(0,9),sigma_2=rep(0,9),sigma_4=rep(0,9))
test_rmse_mat = cbind(sigma_1=rep(0,9),sigma_2=rep(0,9),sigma_4=rep(0,9))

#model counter variable to increase the index of the column in the matrix
model1_counter_all_sigma = 0 
model2_counter_all_sigma = 0 
model3_counter_all_sigma = 0 
model4_counter_all_sigma = 0 
model5_counter_all_sigma = 0 
model6_counter_all_sigma = 0 
model7_counter_all_sigma = 0 
model8_counter_all_sigma = 0 
model9_counter_all_sigma = 0 
  
#Looping through the each sigma to simulate for each 9 models
for(s in 1:length(sigma)){
  for(i in 1:no_loop){
    #epsilon = rnorm(n,mean = 0, sd = sigma[s])
    mlr_data=mlr_sim(sig_data,sd=sigma[s])
    
    #Split the Data into Train Data (Random)
    trn_idx = sample(1:nrow(mlr_data), 250)
    n_test = nrow(mlr_data) - length(trn_idx)

    #Model Creation    
    model1 = lm(y~x1,data=mlr_data[trn_idx,])
    model2 = lm(y~x1+x2,data=mlr_data[trn_idx,])
    model3 = lm(y~x1+x2+x3,data=mlr_data[trn_idx,])
    model4 = lm(y~x1+x2+x3+x4,data=mlr_data[trn_idx,])
    model5 = lm(y~x1+x2+x3+x4+x5,data=mlr_data[trn_idx,])
    model6 = lm(y~x1+x2+x3+x4+x5+x6,data=mlr_data[trn_idx,])
    model7 = lm(y~x1+x2+x3+x4+x5+x6+x7,data=mlr_data[trn_idx,])
    model8 = lm(y~x1+x2+x3+x4+x5+x6+x7+x8,data=mlr_data[trn_idx,])
    model9 = lm(y~x1+x2+x3+x4+x5+x6+x7+x8+x9,data=mlr_data[trn_idx,])
    
    #Predict the Test Data for th 9 model
    newdata_model1 = subset(mlr_data[-trn_idx,],select=c("x1"))
    newdata_model2 = subset(mlr_data[-trn_idx,],select=c("x1","x2"))
    newdata_model3 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3"))
    newdata_model4 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4"))
    newdata_model5 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5"))
    newdata_model6 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6"))
    newdata_model7 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7"))
    newdata_model8 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7","x8"))
    newdata_model9 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7","x8","x9"))    
    
    #Train RMSE
    train_rmse_model1 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model1))^2) / length(trn_idx))
    train_rmse_model2 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model2))^2) / length(trn_idx))
    train_rmse_model3 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model3))^2) / length(trn_idx))
    train_rmse_model4 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model4))^2) / length(trn_idx))
    train_rmse_model5 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model5))^2) / length(trn_idx))
    train_rmse_model6 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model6))^2) / length(trn_idx))
    train_rmse_model7 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model7))^2) / length(trn_idx))
    train_rmse_model8 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model8))^2) / length(trn_idx))
    train_rmse_model9 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model9))^2) / length(trn_idx))
    train_rmse_model1_mat[i,s] = train_rmse_model1
    train_rmse_model2_mat[i,s] = train_rmse_model2
    train_rmse_model3_mat[i,s] = train_rmse_model3
    train_rmse_model4_mat[i,s] = train_rmse_model4
    train_rmse_model5_mat[i,s] = train_rmse_model5
    train_rmse_model6_mat[i,s] = train_rmse_model6
    train_rmse_model7_mat[i,s] = train_rmse_model7
    train_rmse_model8_mat[i,s] = train_rmse_model8
    train_rmse_model9_mat[i,s] = train_rmse_model9    
    
    #Test RMSE    
    test_rmse_model1 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model1,newdata=newdata_model1))^2) / n_test)
    test_rmse_model2 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model2,newdata=newdata_model2))^2) / n_test)
    test_rmse_model3 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model3,newdata=newdata_model3))^2) / n_test)
    test_rmse_model4 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model4,newdata=newdata_model4))^2) / n_test)
    test_rmse_model5 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model5,newdata=newdata_model5))^2) / n_test)
    test_rmse_model6 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model6,newdata=newdata_model6))^2) / n_test)
    test_rmse_model7 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model7,newdata=newdata_model7))^2) / n_test)
    test_rmse_model8 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model8,newdata=newdata_model8))^2) / n_test)
    test_rmse_model9 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model9,newdata=newdata_model9))^2) / n_test)
    test_rmse_model1_mat[i,s] = test_rmse_model1
    test_rmse_model2_mat[i,s] = test_rmse_model2
    test_rmse_model3_mat[i,s] = test_rmse_model3
    test_rmse_model4_mat[i,s] = test_rmse_model4
    test_rmse_model5_mat[i,s] = test_rmse_model5
    test_rmse_model6_mat[i,s] = test_rmse_model6
    test_rmse_model7_mat[i,s] = test_rmse_model7
    test_rmse_model8_mat[i,s] = test_rmse_model8
    test_rmse_model9_mat[i,s] = test_rmse_model9

    #Increase the Model Counter
    model1_counter_all_sigma = model1_counter_all_sigma + 1
    model2_counter_all_sigma = model2_counter_all_sigma + 1
    model3_counter_all_sigma = model3_counter_all_sigma + 1 
    model4_counter_all_sigma = model4_counter_all_sigma + 1 
    model5_counter_all_sigma = model5_counter_all_sigma + 1
    model6_counter_all_sigma = model6_counter_all_sigma + 1 
    model7_counter_all_sigma = model7_counter_all_sigma + 1
    model8_counter_all_sigma = model8_counter_all_sigma + 1
    model9_counter_all_sigma = model9_counter_all_sigma + 1
  }
}

#Print the message of Number of times the each model (1/2/3/4/5/6/7/8/9) is being trained for each value of sigma
print (paste("Number of times the model : 1 trained for alpha:",sigma[s]," is: ",model1_counter_all_sigma))
## [1] "Number of times the model : 1 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 2 trained for alpha:",sigma[s]," is: ",model2_counter_all_sigma))
## [1] "Number of times the model : 2 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 3 trained for alpha:",sigma[s]," is: ",model3_counter_all_sigma))
## [1] "Number of times the model : 3 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 4 trained for alpha:",sigma[s]," is: ",model4_counter_all_sigma))
## [1] "Number of times the model : 4 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 5 trained for alpha:",sigma[s]," is: ",model5_counter_all_sigma))
## [1] "Number of times the model : 5 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 6 trained for alpha:",sigma[s]," is: ",model6_counter_all_sigma))  
## [1] "Number of times the model : 6 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 7 trained for alpha:",sigma[s]," is: ",model7_counter_all_sigma))  
## [1] "Number of times the model : 7 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 8 trained for alpha:",sigma[s]," is: ",model8_counter_all_sigma))  
## [1] "Number of times the model : 8 trained for alpha: 4  is:  3000"
print (paste("Number of times the model : 9 trained for alpha:",sigma[s]," is: ",model9_counter_all_sigma))  
## [1] "Number of times the model : 9 trained for alpha: 4  is:  3000"
#Average out the Train RMSE / Test RMSE for each value of sigma
for(s in 1:length(sigma)){
  train_rmse_mat[1,s] = mean(train_rmse_model1_mat[,s])
  train_rmse_mat[2,s] = mean(train_rmse_model2_mat[,s])
  train_rmse_mat[3,s] = mean(train_rmse_model3_mat[,s])
  train_rmse_mat[4,s] = mean(train_rmse_model4_mat[,s])
  train_rmse_mat[5,s] = mean(train_rmse_model5_mat[,s])
  train_rmse_mat[6,s] = mean(train_rmse_model6_mat[,s])
  train_rmse_mat[7,s] = mean(train_rmse_model7_mat[,s])
  train_rmse_mat[8,s] = mean(train_rmse_model8_mat[,s])
  train_rmse_mat[9,s] = mean(train_rmse_model9_mat[,s])  
  
  test_rmse_mat[1,s] = mean(test_rmse_model1_mat[,s])
  test_rmse_mat[2,s] = mean(test_rmse_model2_mat[,s])
  test_rmse_mat[3,s] = mean(test_rmse_model3_mat[,s])
  test_rmse_mat[4,s] = mean(test_rmse_model4_mat[,s])
  test_rmse_mat[5,s] = mean(test_rmse_model5_mat[,s])
  test_rmse_mat[6,s] = mean(test_rmse_model6_mat[,s])
  test_rmse_mat[7,s] = mean(test_rmse_model7_mat[,s])
  test_rmse_mat[8,s] = mean(test_rmse_model8_mat[,s])
  test_rmse_mat[9,s] = mean(test_rmse_model9_mat[,s])
}

[2.3] Results

As we have discussed in the Introduction on the aspect of Goal, we have to producded plots to support these Goals

  • Simulate the MLR models for 1000 times for a given value of σ
  • Plot 3 curves, each plot is against each sigma, shows what the Train RMSE and Test RMSE against the Model Size (Model1,2,3,4,5,6).

[2.3.1] Train/Test RMSE against 3 sigma values ((1,2,4), for each model (1/2/3…/7/8/9)

plot(seq(1,9), train_rmse_mat[,1], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 1",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,1], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,1], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)

The above plot shows the Train RMSE and Test RMSE for sigma = 1, is being very close to each other. As the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, the test RMSE increases after the Model 6

plot(seq(1,9), train_rmse_mat[,2], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 2",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,2], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,2], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)

The above plot shows the Train RMSE and Test RMSE for sigma = 2, is being very close to each other till model 6, after model 6 they are ditant apart to each other. Also As the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, after Model 6, the Test RMSE increases

plot(seq(1,9), train_rmse_mat[,3], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 4",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,3], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,3], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)

The above plot shows the Train RMSE and Test RMSE for sigma = 4, is being very close to each other till model 2, after model 2 they are distant apart to each other and distance between them increase as the model grows. Also as the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, after the Model 6, the Test RMSE value increases

[2.4] Discussion

We will discuss on the following points, as we mentioned in the Introduction section

  • Does the simulation method always select the correct model (i.e. model 6), or sometimes it select the other models
  • If we do the average of the Test RMSE, whether the simulation method selects the correct model (i.e. Model 6)
  • How does the level of noise (σ = 1,2 and 4) affects the Model selection.

[2.4.1] Does the simulation method always select the correct model?

plot(test_rmse_model1_mat[,1],col="dodgerblue",ylim=(c(0.8,3)),pch=1,lty=1,xlab = "Index", ylab = "RMSE")
points(test_rmse_model2_mat[,1], col="orangered",ylim=(c(0.8,3)),pch=1,lty=2)
points(test_rmse_model3_mat[,1], col="blue4",ylim=(c(0.8,3)),pch=1,lty=3)
points(test_rmse_model4_mat[,1], col="limegreen",ylim=(c(0.8,3)),pch=1,lty=3)
points(test_rmse_model5_mat[,1], col="orange",ylim=(c(0.8,3)),pch=1,lty=4)
points(test_rmse_model6_mat[,1], col="darkorchid",ylim=(c(0.8,3)),pch=6,lty=5)
points(test_rmse_model7_mat[,1], col="violetred",ylim=(c(0.8,3)),pch=1,lty=6)
points(test_rmse_model8_mat[,1], col="tan3",ylim=(c(0.8,3)),pch=1,lty=7)
points(test_rmse_model9_mat[,1], col="deeppink",ylim=(c(0.8,3)),pch=1,lty=8)
legend("topright",legend=c("Model1","Model2","Model3","Model4","Model5","Model6","Model7","Model8","Model9"),col=c("dodgerblue","orangered","blue4","limegreen","orange","darkorchid","violetred","tan3","deeppink"),lty=c(1,2,3,4,5,6,7,8,9))

The above plots the RMSE of all model created from the simulation for the Sigma = 1

The correct model should be the one which should have the lowest RMSE. From the above plots (which plots against all the 1000 RMSE of simulation) it shows that the Model 1(blue color) and Model 2(red color) have higher RMSE and distinctly far from the other model (Model 3/4/5/6/7/8/9), lets ignore the Model 1 and Model 2. Let’s explore the other models as below

plot(test_rmse_model3_mat[,1], col="dodgerblue",ylim=(c(0.9,1.3)),pch=1,lty=3,xlab="Indx",ylab="RMSE")
points(test_rmse_model4_mat[,1], col="limegreen",ylim=(c(0.9,1.3)),pch=1,lty=3)
points(test_rmse_model5_mat[,1], col="orange",ylim=(c(0.9,1.3)),pch=1,lty=4)
points(test_rmse_model6_mat[,1], col="darkorchid",ylim=(c(0.9,1.3)),pch=8,lty=5)
points(test_rmse_model7_mat[,1], col="violetred",ylim=(c(0.9,1.3)),pch=1,lty=6)
points(test_rmse_model8_mat[,1], col="tan3",ylim=(c(0.9,1.3)),pch=1,lty=7)
points(test_rmse_model9_mat[,1], col="deeppink",ylim=(c(0.9,1.3)),pch=1,lty=8)
legend("topright",legend=c("Model3","Model4","Model5","Model6","Model7","Model8","Model9"),col=c("blue4","limegreen","orange","darkorchid","violetred","tan3","deeppink"),pch=c(1,1,1,8,1,1),lty=c(3,4,5,6,7,8,9))

The above model plots the RMSE of models (3/4/5/6/7/8/9) created from the simulation for the Sigma = 1

From the above plot, its clear that the Model 6 not always (marked in darkorchid) selected as the best model, because doesn’t shows the RMSE is lowest among the other models(2/3/4/5/7/8/9). There are many instances, where the Model 6 RMSE is higher than that of other models. Also though, there are instances where the Model 6 RMSE is the lowest among other models. However other models also have the RMSE in range with the model 6. Hence, if we consider all the 1000 simulated RMSE, Model 6 is not the best model always being selected. Let’s take a loot at the average plot of RMSE of all the models and plot

[2.4.4] Average of the Test RMSE, methods selects the correct model (i.e. Model 6)?

plot(seq(1,9), test_rmse_mat[,1], type="o", col="chartreuse4", pch=8, lty=1,main="Sigma = 1",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,1]), col="chartreuse4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 1

The above plot shows the model 6 has the lowest RMSE which is around 1.0148456, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model

plot(seq(1,9), test_rmse_mat[,2], type="o", col="blue4", pch="*", lty=1,main="Sigma = 2",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,2]), col="blue4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 2

The above plot shows the model 6 has the lowest RMSE which is around 2.030618, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model

plot(seq(1,9), test_rmse_mat[,3], type="o", col="red", pch="*", lty=1,main="Sigma = 4",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,3]), col="red",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 4

The above plot shows the model 6 has the lowest RMSE which is around 4.0538772, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model

plot(seq(1,9), test_rmse_mat[,3], col="red",lty=1,ylim=c(0.5,5),xlab="MODEL SIZE",ylab="RMSE")
lines(seq(1,9), test_rmse_mat[,3], col="red",lty=1)
points(6, min(test_rmse_mat[,3]), col="red",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 4
points(seq(1,9), test_rmse_mat[,2], col="blue4",lty=2)
lines(seq(1,9), test_rmse_mat[,2], col="blue4",lty=2)
points(6, min(test_rmse_mat[,2]), col="blue4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 2
points(seq(1,9), test_rmse_mat[,1], col="chartreuse4",lty=3)
lines(seq(1,9), test_rmse_mat[,1], col="chartreuse4",lty=3)
points(6, min(test_rmse_mat[,1]), col="chartreuse4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 1
legend("topright",legend=c("sigma = 4","sigma = 2","simga = 1"),col=c("red","blue4","chartreuse4"),lty=c(1,2,3))

The above plots shows the comparision of the average RMSE of all the 9 models for respective 3 sigma values (1, 2 and 4). The big filled circle points shows the lowest of the RMSE against the Model Size. From the comparision of the plot, it shows that lowest RMSE is from the sigma =1, also noted that higher the value of sigma, higher will be the Test RMSE. Hence to lower the RMSE for the test, we need to choose the lower value of noise (i.e. sigma)

[3] Simulation Study 3, Power

[3.1] Introduction

In this simulation project we will investigate the power of the signifincance of the regression test for the simple linear regression (SLR) model

\(H_0 : \beta_{0} = 0 Vs H_1: \beta_{1} \neq 0\)

Where, Power is the probability of rejecting the null hypothesis when the null is not true, that is, the alternative is true and \(\beta_{1}\) is non-zero.

We will do the simulation of the Simple linear regression (SLR) of the model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \epsilon_i.\].

For the simplicity we will make \(\beta_0\) = 0, thus \(\beta_1\) is essentially controlling the amount of signal. We will be considering the following for the different signals , noises and sample sizes

  • β1 ∈ (−2,−1.9,−1.8,…,−0.1,0,0.1,0.2,0.3,…1.9,2) , which is 41 values
  • σ ∈ (1,2,4), three distinct values of sigma
  • n ∈(10,20,30), three distinct values of n (sample size)

We will hold the value if the \(\alpha\) as 0.05, while validing the hypothesis. For the generation of the sample data, will be using the seq(0, 5, length = n) for different values of the sample size (i.e., n)

For each possible combination of \(\beta_{1}\) and \(\alpha\), we will simulate 1000 times to perform the significance of linear regression (SLR) and following formula will be using to measure the Power

\(\hat{Power}\) = \(\hat{P}\)[Reject \(H_0\), \(H_1\) True] = \(\dfrac{\#Test Rejected}{\#Simulations}\)

[3.1.1] Goal

As part of this exercise, the following are the Goals to achieve

  • Simulate the SLR models for 1000 times for a given value of σ, n , β1 and stored the Power you get
  • Plot the Power Curve against the signal strength (β1) for different value of n, plot this kind of plot for each value of σ
  • Discuss with relevant plots, what are the impact of σ, n , β1 on the Power, as following
    • What’s the impact of σ on Power
    • What’s the impact of n on Power
    • What’s the impact of β1 on Power
  • As we have simulated for the 1000 iterations, would it be suffice ? or more simulations have more important informations to revealed?

In order to achieve these goals, we have various methods in the form of R codes, markdown

[3.2] Methods

Below are the methods proposed which will runs the 1000 simulations against β1, σ and n. After each simulations,

  • The power value is being agregated and stored in the matrix model_sim
  • The 1st column of the row of the matix is the β1 value
  • 2nd, 3rd and 4th column of matrix = model_sim, stores the power values for sigma = 1, for the sample size 10, 20 and 30 respectively
  • 5th, 6th and 7th column of matrix = model_sim, stores the power values for sigma = 2, for the sample size of 10, 20 and 30 respectively
  • 8th, 9th and 10th column of matrix = model_sim, stores the power values for sigma = 5, for the sample size of 10, 20 and 30 respectively

Once the matrix model_sim is populated, we will be using the matix to have different plots to address different discussion

[3.2.1] Code for Simulations

birthday = 19770411
set.seed(birthday)

alpha = 0.05
beta_1 = seq(from=-2,to=2,by=0.1)
sigma = c(1,2,4)
n = c(10,20,30)
no_loop = 1000

#simple linear regression(SLR) Function to create sample data
slr_sim = function(n_in,sd=1,beta_1_in=-2){
  x_values = seq(0, 5, length = n_in)
  epsilon = rnorm(n_in,mean = 0,sd = sd)
  y = beta_1_in * x_values + epsilon
  data.frame(predictor = x_values,response=y)
}

#Matrix to store the Power of the simulated models
model_sim = cbind(beta_1_sim_val=rep(0,length(beta_1)),n_10_sigma_1=rep(0,length(beta_1)),n_20_sigma_1=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)),n_10_sigma_2=rep(0,length(beta_1)),n_20_sigma_2=rep(0,length(beta_1)),n_30_sigma_2=rep(0,length(beta_1)),n_10_sigma_4=rep(0,length(beta_1)),n_20_sigma_4=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)))

model_sim[,1] = seq(from=-2,to=2,by=0.1)
extra_counter = 0

#Loop to iterate over Sigma, number of samples, beta_1 values and 1000 simulations to extract and store the Power values
run_counter = 0
for(s in 1:length(sigma)){ #sigma = c(1,2,4)
  for (n_s in 1:length(n)){ #n = c(10,20,30)
    for (b in 1:length(beta_1)){ # beta_1 = seq(from=-2,to=2,by=0.1)
      tot_sum_signifincace_sim = 0
      run_counter = run_counter + 1      
      for(i in 1:no_loop){
        slr_data = slr_sim(n_in = n[n_s], sd = sigma[s], beta_1_in = beta_1[b] )
        model = lm(response~predictor, data = slr_data)
        model_p_val = summary(model)$coefficient[2,4]
        tot_sum_signifincace_sim = tot_sum_signifincace_sim + ifelse(model_p_val<alpha,1,0)
      }
      power_sim_beta = tot_sum_signifincace_sim / no_loop
      model_sim[b,s+n_s+extra_counter] = power_sim_beta    
      print (paste(run_counter," Sigma:",sigma[s]," n:",n[n_s]," beta_1:",beta_1[b]," Signifiance_Sum:",tot_sum_signifincace_sim, " Power:",power_sim_beta))
    }
  }
  extra_counter = extra_counter + 2
}
## [1] "1  Sigma: 1  n: 10  beta_1: -2  Signifiance_Sum: 1000  Power: 1"
## [1] "2  Sigma: 1  n: 10  beta_1: -1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "3  Sigma: 1  n: 10  beta_1: -1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "4  Sigma: 1  n: 10  beta_1: -1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "5  Sigma: 1  n: 10  beta_1: -1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "6  Sigma: 1  n: 10  beta_1: -1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "7  Sigma: 1  n: 10  beta_1: -1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "8  Sigma: 1  n: 10  beta_1: -1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "9  Sigma: 1  n: 10  beta_1: -1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "10  Sigma: 1  n: 10  beta_1: -1.1  Signifiance_Sum: 998  Power: 0.998"
## [1] "11  Sigma: 1  n: 10  beta_1: -1  Signifiance_Sum: 992  Power: 0.992"
## [1] "12  Sigma: 1  n: 10  beta_1: -0.9  Signifiance_Sum: 980  Power: 0.98"
## [1] "13  Sigma: 1  n: 10  beta_1: -0.8  Signifiance_Sum: 935  Power: 0.935"
## [1] "14  Sigma: 1  n: 10  beta_1: -0.7  Signifiance_Sum: 880  Power: 0.88"
## [1] "15  Sigma: 1  n: 10  beta_1: -0.6  Signifiance_Sum: 743  Power: 0.743"
## [1] "16  Sigma: 1  n: 10  beta_1: -0.5  Signifiance_Sum: 599  Power: 0.599"
## [1] "17  Sigma: 1  n: 10  beta_1: -0.4  Signifiance_Sum: 430  Power: 0.43"
## [1] "18  Sigma: 1  n: 10  beta_1: -0.3  Signifiance_Sum: 270  Power: 0.27"
## [1] "19  Sigma: 1  n: 10  beta_1: -0.2  Signifiance_Sum: 130  Power: 0.13"
## [1] "20  Sigma: 1  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 84  Power: 0.084"
## [1] "21  Sigma: 1  n: 10  beta_1: 0  Signifiance_Sum: 41  Power: 0.041"
## [1] "22  Sigma: 1  n: 10  beta_1: 0.1  Signifiance_Sum: 68  Power: 0.068"
## [1] "23  Sigma: 1  n: 10  beta_1: 0.2  Signifiance_Sum: 152  Power: 0.152"
## [1] "24  Sigma: 1  n: 10  beta_1: 0.3  Signifiance_Sum: 256  Power: 0.256"
## [1] "25  Sigma: 1  n: 10  beta_1: 0.4  Signifiance_Sum: 416  Power: 0.416"
## [1] "26  Sigma: 1  n: 10  beta_1: 0.5  Signifiance_Sum: 584  Power: 0.584"
## [1] "27  Sigma: 1  n: 10  beta_1: 0.6  Signifiance_Sum: 713  Power: 0.713"
## [1] "28  Sigma: 1  n: 10  beta_1: 0.7  Signifiance_Sum: 868  Power: 0.868"
## [1] "29  Sigma: 1  n: 10  beta_1: 0.8  Signifiance_Sum: 942  Power: 0.942"
## [1] "30  Sigma: 1  n: 10  beta_1: 0.9  Signifiance_Sum: 971  Power: 0.971"
## [1] "31  Sigma: 1  n: 10  beta_1: 1  Signifiance_Sum: 992  Power: 0.992"
## [1] "32  Sigma: 1  n: 10  beta_1: 1.1  Signifiance_Sum: 998  Power: 0.998"
## [1] "33  Sigma: 1  n: 10  beta_1: 1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "34  Sigma: 1  n: 10  beta_1: 1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "35  Sigma: 1  n: 10  beta_1: 1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "36  Sigma: 1  n: 10  beta_1: 1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "37  Sigma: 1  n: 10  beta_1: 1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "38  Sigma: 1  n: 10  beta_1: 1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "39  Sigma: 1  n: 10  beta_1: 1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "40  Sigma: 1  n: 10  beta_1: 1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "41  Sigma: 1  n: 10  beta_1: 2  Signifiance_Sum: 1000  Power: 1"
## [1] "42  Sigma: 1  n: 20  beta_1: -2  Signifiance_Sum: 1000  Power: 1"
## [1] "43  Sigma: 1  n: 20  beta_1: -1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "44  Sigma: 1  n: 20  beta_1: -1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "45  Sigma: 1  n: 20  beta_1: -1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "46  Sigma: 1  n: 20  beta_1: -1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "47  Sigma: 1  n: 20  beta_1: -1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "48  Sigma: 1  n: 20  beta_1: -1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "49  Sigma: 1  n: 20  beta_1: -1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "50  Sigma: 1  n: 20  beta_1: -1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "51  Sigma: 1  n: 20  beta_1: -1.1  Signifiance_Sum: 1000  Power: 1"
## [1] "52  Sigma: 1  n: 20  beta_1: -1  Signifiance_Sum: 1000  Power: 1"
## [1] "53  Sigma: 1  n: 20  beta_1: -0.9  Signifiance_Sum: 1000  Power: 1"
## [1] "54  Sigma: 1  n: 20  beta_1: -0.8  Signifiance_Sum: 1000  Power: 1"
## [1] "55  Sigma: 1  n: 20  beta_1: -0.7  Signifiance_Sum: 990  Power: 0.99"
## [1] "56  Sigma: 1  n: 20  beta_1: -0.6  Signifiance_Sum: 967  Power: 0.967"
## [1] "57  Sigma: 1  n: 20  beta_1: -0.5  Signifiance_Sum: 897  Power: 0.897"
## [1] "58  Sigma: 1  n: 20  beta_1: -0.4  Signifiance_Sum: 745  Power: 0.745"
## [1] "59  Sigma: 1  n: 20  beta_1: -0.3  Signifiance_Sum: 513  Power: 0.513"
## [1] "60  Sigma: 1  n: 20  beta_1: -0.2  Signifiance_Sum: 234  Power: 0.234"
## [1] "61  Sigma: 1  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 116  Power: 0.116"
## [1] "62  Sigma: 1  n: 20  beta_1: 0  Signifiance_Sum: 40  Power: 0.04"
## [1] "63  Sigma: 1  n: 20  beta_1: 0.1  Signifiance_Sum: 108  Power: 0.108"
## [1] "64  Sigma: 1  n: 20  beta_1: 0.2  Signifiance_Sum: 237  Power: 0.237"
## [1] "65  Sigma: 1  n: 20  beta_1: 0.3  Signifiance_Sum: 500  Power: 0.5"
## [1] "66  Sigma: 1  n: 20  beta_1: 0.4  Signifiance_Sum: 727  Power: 0.727"
## [1] "67  Sigma: 1  n: 20  beta_1: 0.5  Signifiance_Sum: 908  Power: 0.908"
## [1] "68  Sigma: 1  n: 20  beta_1: 0.6  Signifiance_Sum: 966  Power: 0.966"
## [1] "69  Sigma: 1  n: 20  beta_1: 0.7  Signifiance_Sum: 999  Power: 0.999"
## [1] "70  Sigma: 1  n: 20  beta_1: 0.8  Signifiance_Sum: 998  Power: 0.998"
## [1] "71  Sigma: 1  n: 20  beta_1: 0.9  Signifiance_Sum: 1000  Power: 1"
## [1] "72  Sigma: 1  n: 20  beta_1: 1  Signifiance_Sum: 1000  Power: 1"
## [1] "73  Sigma: 1  n: 20  beta_1: 1.1  Signifiance_Sum: 1000  Power: 1"
## [1] "74  Sigma: 1  n: 20  beta_1: 1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "75  Sigma: 1  n: 20  beta_1: 1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "76  Sigma: 1  n: 20  beta_1: 1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "77  Sigma: 1  n: 20  beta_1: 1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "78  Sigma: 1  n: 20  beta_1: 1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "79  Sigma: 1  n: 20  beta_1: 1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "80  Sigma: 1  n: 20  beta_1: 1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "81  Sigma: 1  n: 20  beta_1: 1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "82  Sigma: 1  n: 20  beta_1: 2  Signifiance_Sum: 1000  Power: 1"
## [1] "83  Sigma: 1  n: 30  beta_1: -2  Signifiance_Sum: 1000  Power: 1"
## [1] "84  Sigma: 1  n: 30  beta_1: -1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "85  Sigma: 1  n: 30  beta_1: -1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "86  Sigma: 1  n: 30  beta_1: -1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "87  Sigma: 1  n: 30  beta_1: -1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "88  Sigma: 1  n: 30  beta_1: -1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "89  Sigma: 1  n: 30  beta_1: -1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "90  Sigma: 1  n: 30  beta_1: -1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "91  Sigma: 1  n: 30  beta_1: -1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "92  Sigma: 1  n: 30  beta_1: -1.1  Signifiance_Sum: 1000  Power: 1"
## [1] "93  Sigma: 1  n: 30  beta_1: -1  Signifiance_Sum: 1000  Power: 1"
## [1] "94  Sigma: 1  n: 30  beta_1: -0.9  Signifiance_Sum: 1000  Power: 1"
## [1] "95  Sigma: 1  n: 30  beta_1: -0.8  Signifiance_Sum: 1000  Power: 1"
## [1] "96  Sigma: 1  n: 30  beta_1: -0.7  Signifiance_Sum: 999  Power: 0.999"
## [1] "97  Sigma: 1  n: 30  beta_1: -0.6  Signifiance_Sum: 994  Power: 0.994"
## [1] "98  Sigma: 1  n: 30  beta_1: -0.5  Signifiance_Sum: 972  Power: 0.972"
## [1] "99  Sigma: 1  n: 30  beta_1: -0.4  Signifiance_Sum: 880  Power: 0.88"
## [1] "100  Sigma: 1  n: 30  beta_1: -0.3  Signifiance_Sum: 683  Power: 0.683"
## [1] "101  Sigma: 1  n: 30  beta_1: -0.2  Signifiance_Sum: 371  Power: 0.371"
## [1] "102  Sigma: 1  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 130  Power: 0.13"
## [1] "103  Sigma: 1  n: 30  beta_1: 0  Signifiance_Sum: 41  Power: 0.041"
## [1] "104  Sigma: 1  n: 30  beta_1: 0.1  Signifiance_Sum: 127  Power: 0.127"
## [1] "105  Sigma: 1  n: 30  beta_1: 0.2  Signifiance_Sum: 345  Power: 0.345"
## [1] "106  Sigma: 1  n: 30  beta_1: 0.3  Signifiance_Sum: 664  Power: 0.664"
## [1] "107  Sigma: 1  n: 30  beta_1: 0.4  Signifiance_Sum: 889  Power: 0.889"
## [1] "108  Sigma: 1  n: 30  beta_1: 0.5  Signifiance_Sum: 983  Power: 0.983"
## [1] "109  Sigma: 1  n: 30  beta_1: 0.6  Signifiance_Sum: 996  Power: 0.996"
## [1] "110  Sigma: 1  n: 30  beta_1: 0.7  Signifiance_Sum: 1000  Power: 1"
## [1] "111  Sigma: 1  n: 30  beta_1: 0.8  Signifiance_Sum: 1000  Power: 1"
## [1] "112  Sigma: 1  n: 30  beta_1: 0.9  Signifiance_Sum: 1000  Power: 1"
## [1] "113  Sigma: 1  n: 30  beta_1: 1  Signifiance_Sum: 1000  Power: 1"
## [1] "114  Sigma: 1  n: 30  beta_1: 1.1  Signifiance_Sum: 1000  Power: 1"
## [1] "115  Sigma: 1  n: 30  beta_1: 1.2  Signifiance_Sum: 1000  Power: 1"
## [1] "116  Sigma: 1  n: 30  beta_1: 1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "117  Sigma: 1  n: 30  beta_1: 1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "118  Sigma: 1  n: 30  beta_1: 1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "119  Sigma: 1  n: 30  beta_1: 1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "120  Sigma: 1  n: 30  beta_1: 1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "121  Sigma: 1  n: 30  beta_1: 1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "122  Sigma: 1  n: 30  beta_1: 1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "123  Sigma: 1  n: 30  beta_1: 2  Signifiance_Sum: 1000  Power: 1"
## [1] "124  Sigma: 2  n: 10  beta_1: -2  Signifiance_Sum: 991  Power: 0.991"
## [1] "125  Sigma: 2  n: 10  beta_1: -1.9  Signifiance_Sum: 990  Power: 0.99"
## [1] "126  Sigma: 2  n: 10  beta_1: -1.8  Signifiance_Sum: 975  Power: 0.975"
## [1] "127  Sigma: 2  n: 10  beta_1: -1.7  Signifiance_Sum: 957  Power: 0.957"
## [1] "128  Sigma: 2  n: 10  beta_1: -1.6  Signifiance_Sum: 942  Power: 0.942"
## [1] "129  Sigma: 2  n: 10  beta_1: -1.5  Signifiance_Sum: 906  Power: 0.906"
## [1] "130  Sigma: 2  n: 10  beta_1: -1.4  Signifiance_Sum: 887  Power: 0.887"
## [1] "131  Sigma: 2  n: 10  beta_1: -1.3  Signifiance_Sum: 831  Power: 0.831"
## [1] "132  Sigma: 2  n: 10  beta_1: -1.2  Signifiance_Sum: 742  Power: 0.742"
## [1] "133  Sigma: 2  n: 10  beta_1: -1.1  Signifiance_Sum: 676  Power: 0.676"
## [1] "134  Sigma: 2  n: 10  beta_1: -1  Signifiance_Sum: 587  Power: 0.587"
## [1] "135  Sigma: 2  n: 10  beta_1: -0.9  Signifiance_Sum: 562  Power: 0.562"
## [1] "136  Sigma: 2  n: 10  beta_1: -0.8  Signifiance_Sum: 404  Power: 0.404"
## [1] "137  Sigma: 2  n: 10  beta_1: -0.7  Signifiance_Sum: 347  Power: 0.347"
## [1] "138  Sigma: 2  n: 10  beta_1: -0.6  Signifiance_Sum: 263  Power: 0.263"
## [1] "139  Sigma: 2  n: 10  beta_1: -0.5  Signifiance_Sum: 199  Power: 0.199"
## [1] "140  Sigma: 2  n: 10  beta_1: -0.4  Signifiance_Sum: 140  Power: 0.14"
## [1] "141  Sigma: 2  n: 10  beta_1: -0.3  Signifiance_Sum: 108  Power: 0.108"
## [1] "142  Sigma: 2  n: 10  beta_1: -0.2  Signifiance_Sum: 63  Power: 0.063"
## [1] "143  Sigma: 2  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 56  Power: 0.056"
## [1] "144  Sigma: 2  n: 10  beta_1: 0  Signifiance_Sum: 44  Power: 0.044"
## [1] "145  Sigma: 2  n: 10  beta_1: 0.1  Signifiance_Sum: 60  Power: 0.06"
## [1] "146  Sigma: 2  n: 10  beta_1: 0.2  Signifiance_Sum: 77  Power: 0.077"
## [1] "147  Sigma: 2  n: 10  beta_1: 0.3  Signifiance_Sum: 98  Power: 0.098"
## [1] "148  Sigma: 2  n: 10  beta_1: 0.4  Signifiance_Sum: 158  Power: 0.158"
## [1] "149  Sigma: 2  n: 10  beta_1: 0.5  Signifiance_Sum: 214  Power: 0.214"
## [1] "150  Sigma: 2  n: 10  beta_1: 0.6  Signifiance_Sum: 258  Power: 0.258"
## [1] "151  Sigma: 2  n: 10  beta_1: 0.7  Signifiance_Sum: 338  Power: 0.338"
## [1] "152  Sigma: 2  n: 10  beta_1: 0.8  Signifiance_Sum: 424  Power: 0.424"
## [1] "153  Sigma: 2  n: 10  beta_1: 0.9  Signifiance_Sum: 518  Power: 0.518"
## [1] "154  Sigma: 2  n: 10  beta_1: 1  Signifiance_Sum: 605  Power: 0.605"
## [1] "155  Sigma: 2  n: 10  beta_1: 1.1  Signifiance_Sum: 689  Power: 0.689"
## [1] "156  Sigma: 2  n: 10  beta_1: 1.2  Signifiance_Sum: 772  Power: 0.772"
## [1] "157  Sigma: 2  n: 10  beta_1: 1.3  Signifiance_Sum: 829  Power: 0.829"
## [1] "158  Sigma: 2  n: 10  beta_1: 1.4  Signifiance_Sum: 887  Power: 0.887"
## [1] "159  Sigma: 2  n: 10  beta_1: 1.5  Signifiance_Sum: 922  Power: 0.922"
## [1] "160  Sigma: 2  n: 10  beta_1: 1.6  Signifiance_Sum: 948  Power: 0.948"
## [1] "161  Sigma: 2  n: 10  beta_1: 1.7  Signifiance_Sum: 969  Power: 0.969"
## [1] "162  Sigma: 2  n: 10  beta_1: 1.8  Signifiance_Sum: 974  Power: 0.974"
## [1] "163  Sigma: 2  n: 10  beta_1: 1.9  Signifiance_Sum: 983  Power: 0.983"
## [1] "164  Sigma: 2  n: 10  beta_1: 2  Signifiance_Sum: 996  Power: 0.996"
## [1] "165  Sigma: 2  n: 20  beta_1: -2  Signifiance_Sum: 1000  Power: 1"
## [1] "166  Sigma: 2  n: 20  beta_1: -1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "167  Sigma: 2  n: 20  beta_1: -1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "168  Sigma: 2  n: 20  beta_1: -1.7  Signifiance_Sum: 999  Power: 0.999"
## [1] "169  Sigma: 2  n: 20  beta_1: -1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "170  Sigma: 2  n: 20  beta_1: -1.5  Signifiance_Sum: 999  Power: 0.999"
## [1] "171  Sigma: 2  n: 20  beta_1: -1.4  Signifiance_Sum: 992  Power: 0.992"
## [1] "172  Sigma: 2  n: 20  beta_1: -1.3  Signifiance_Sum: 984  Power: 0.984"
## [1] "173  Sigma: 2  n: 20  beta_1: -1.2  Signifiance_Sum: 967  Power: 0.967"
## [1] "174  Sigma: 2  n: 20  beta_1: -1.1  Signifiance_Sum: 939  Power: 0.939"
## [1] "175  Sigma: 2  n: 20  beta_1: -1  Signifiance_Sum: 890  Power: 0.89"
## [1] "176  Sigma: 2  n: 20  beta_1: -0.9  Signifiance_Sum: 828  Power: 0.828"
## [1] "177  Sigma: 2  n: 20  beta_1: -0.8  Signifiance_Sum: 738  Power: 0.738"
## [1] "178  Sigma: 2  n: 20  beta_1: -0.7  Signifiance_Sum: 601  Power: 0.601"
## [1] "179  Sigma: 2  n: 20  beta_1: -0.6  Signifiance_Sum: 489  Power: 0.489"
## [1] "180  Sigma: 2  n: 20  beta_1: -0.5  Signifiance_Sum: 343  Power: 0.343"
## [1] "181  Sigma: 2  n: 20  beta_1: -0.4  Signifiance_Sum: 254  Power: 0.254"
## [1] "182  Sigma: 2  n: 20  beta_1: -0.3  Signifiance_Sum: 145  Power: 0.145"
## [1] "183  Sigma: 2  n: 20  beta_1: -0.2  Signifiance_Sum: 106  Power: 0.106"
## [1] "184  Sigma: 2  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 64  Power: 0.064"
## [1] "185  Sigma: 2  n: 20  beta_1: 0  Signifiance_Sum: 45  Power: 0.045"
## [1] "186  Sigma: 2  n: 20  beta_1: 0.1  Signifiance_Sum: 62  Power: 0.062"
## [1] "187  Sigma: 2  n: 20  beta_1: 0.2  Signifiance_Sum: 105  Power: 0.105"
## [1] "188  Sigma: 2  n: 20  beta_1: 0.3  Signifiance_Sum: 160  Power: 0.16"
## [1] "189  Sigma: 2  n: 20  beta_1: 0.4  Signifiance_Sum: 232  Power: 0.232"
## [1] "190  Sigma: 2  n: 20  beta_1: 0.5  Signifiance_Sum: 366  Power: 0.366"
## [1] "191  Sigma: 2  n: 20  beta_1: 0.6  Signifiance_Sum: 461  Power: 0.461"
## [1] "192  Sigma: 2  n: 20  beta_1: 0.7  Signifiance_Sum: 602  Power: 0.602"
## [1] "193  Sigma: 2  n: 20  beta_1: 0.8  Signifiance_Sum: 716  Power: 0.716"
## [1] "194  Sigma: 2  n: 20  beta_1: 0.9  Signifiance_Sum: 822  Power: 0.822"
## [1] "195  Sigma: 2  n: 20  beta_1: 1  Signifiance_Sum: 904  Power: 0.904"
## [1] "196  Sigma: 2  n: 20  beta_1: 1.1  Signifiance_Sum: 948  Power: 0.948"
## [1] "197  Sigma: 2  n: 20  beta_1: 1.2  Signifiance_Sum: 968  Power: 0.968"
## [1] "198  Sigma: 2  n: 20  beta_1: 1.3  Signifiance_Sum: 981  Power: 0.981"
## [1] "199  Sigma: 2  n: 20  beta_1: 1.4  Signifiance_Sum: 994  Power: 0.994"
## [1] "200  Sigma: 2  n: 20  beta_1: 1.5  Signifiance_Sum: 997  Power: 0.997"
## [1] "201  Sigma: 2  n: 20  beta_1: 1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "202  Sigma: 2  n: 20  beta_1: 1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "203  Sigma: 2  n: 20  beta_1: 1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "204  Sigma: 2  n: 20  beta_1: 1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "205  Sigma: 2  n: 20  beta_1: 2  Signifiance_Sum: 1000  Power: 1"
## [1] "206  Sigma: 2  n: 30  beta_1: -2  Signifiance_Sum: 1000  Power: 1"
## [1] "207  Sigma: 2  n: 30  beta_1: -1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "208  Sigma: 2  n: 30  beta_1: -1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "209  Sigma: 2  n: 30  beta_1: -1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "210  Sigma: 2  n: 30  beta_1: -1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "211  Sigma: 2  n: 30  beta_1: -1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "212  Sigma: 2  n: 30  beta_1: -1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "213  Sigma: 2  n: 30  beta_1: -1.3  Signifiance_Sum: 1000  Power: 1"
## [1] "214  Sigma: 2  n: 30  beta_1: -1.2  Signifiance_Sum: 998  Power: 0.998"
## [1] "215  Sigma: 2  n: 30  beta_1: -1.1  Signifiance_Sum: 990  Power: 0.99"
## [1] "216  Sigma: 2  n: 30  beta_1: -1  Signifiance_Sum: 973  Power: 0.973"
## [1] "217  Sigma: 2  n: 30  beta_1: -0.9  Signifiance_Sum: 932  Power: 0.932"
## [1] "218  Sigma: 2  n: 30  beta_1: -0.8  Signifiance_Sum: 890  Power: 0.89"
## [1] "219  Sigma: 2  n: 30  beta_1: -0.7  Signifiance_Sum: 790  Power: 0.79"
## [1] "220  Sigma: 2  n: 30  beta_1: -0.6  Signifiance_Sum: 653  Power: 0.653"
## [1] "221  Sigma: 2  n: 30  beta_1: -0.5  Signifiance_Sum: 526  Power: 0.526"
## [1] "222  Sigma: 2  n: 30  beta_1: -0.4  Signifiance_Sum: 343  Power: 0.343"
## [1] "223  Sigma: 2  n: 30  beta_1: -0.3  Signifiance_Sum: 219  Power: 0.219"
## [1] "224  Sigma: 2  n: 30  beta_1: -0.2  Signifiance_Sum: 125  Power: 0.125"
## [1] "225  Sigma: 2  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 53  Power: 0.053"
## [1] "226  Sigma: 2  n: 30  beta_1: 0  Signifiance_Sum: 54  Power: 0.054"
## [1] "227  Sigma: 2  n: 30  beta_1: 0.1  Signifiance_Sum: 67  Power: 0.067"
## [1] "228  Sigma: 2  n: 30  beta_1: 0.2  Signifiance_Sum: 142  Power: 0.142"
## [1] "229  Sigma: 2  n: 30  beta_1: 0.3  Signifiance_Sum: 226  Power: 0.226"
## [1] "230  Sigma: 2  n: 30  beta_1: 0.4  Signifiance_Sum: 345  Power: 0.345"
## [1] "231  Sigma: 2  n: 30  beta_1: 0.5  Signifiance_Sum: 497  Power: 0.497"
## [1] "232  Sigma: 2  n: 30  beta_1: 0.6  Signifiance_Sum: 687  Power: 0.687"
## [1] "233  Sigma: 2  n: 30  beta_1: 0.7  Signifiance_Sum: 784  Power: 0.784"
## [1] "234  Sigma: 2  n: 30  beta_1: 0.8  Signifiance_Sum: 888  Power: 0.888"
## [1] "235  Sigma: 2  n: 30  beta_1: 0.9  Signifiance_Sum: 931  Power: 0.931"
## [1] "236  Sigma: 2  n: 30  beta_1: 1  Signifiance_Sum: 988  Power: 0.988"
## [1] "237  Sigma: 2  n: 30  beta_1: 1.1  Signifiance_Sum: 989  Power: 0.989"
## [1] "238  Sigma: 2  n: 30  beta_1: 1.2  Signifiance_Sum: 997  Power: 0.997"
## [1] "239  Sigma: 2  n: 30  beta_1: 1.3  Signifiance_Sum: 999  Power: 0.999"
## [1] "240  Sigma: 2  n: 30  beta_1: 1.4  Signifiance_Sum: 1000  Power: 1"
## [1] "241  Sigma: 2  n: 30  beta_1: 1.5  Signifiance_Sum: 1000  Power: 1"
## [1] "242  Sigma: 2  n: 30  beta_1: 1.6  Signifiance_Sum: 1000  Power: 1"
## [1] "243  Sigma: 2  n: 30  beta_1: 1.7  Signifiance_Sum: 1000  Power: 1"
## [1] "244  Sigma: 2  n: 30  beta_1: 1.8  Signifiance_Sum: 1000  Power: 1"
## [1] "245  Sigma: 2  n: 30  beta_1: 1.9  Signifiance_Sum: 1000  Power: 1"
## [1] "246  Sigma: 2  n: 30  beta_1: 2  Signifiance_Sum: 1000  Power: 1"
## [1] "247  Sigma: 4  n: 10  beta_1: -2  Signifiance_Sum: 578  Power: 0.578"
## [1] "248  Sigma: 4  n: 10  beta_1: -1.9  Signifiance_Sum: 551  Power: 0.551"
## [1] "249  Sigma: 4  n: 10  beta_1: -1.8  Signifiance_Sum: 520  Power: 0.52"
## [1] "250  Sigma: 4  n: 10  beta_1: -1.7  Signifiance_Sum: 485  Power: 0.485"
## [1] "251  Sigma: 4  n: 10  beta_1: -1.6  Signifiance_Sum: 448  Power: 0.448"
## [1] "252  Sigma: 4  n: 10  beta_1: -1.5  Signifiance_Sum: 390  Power: 0.39"
## [1] "253  Sigma: 4  n: 10  beta_1: -1.4  Signifiance_Sum: 373  Power: 0.373"
## [1] "254  Sigma: 4  n: 10  beta_1: -1.3  Signifiance_Sum: 288  Power: 0.288"
## [1] "255  Sigma: 4  n: 10  beta_1: -1.2  Signifiance_Sum: 284  Power: 0.284"
## [1] "256  Sigma: 4  n: 10  beta_1: -1.1  Signifiance_Sum: 223  Power: 0.223"
## [1] "257  Sigma: 4  n: 10  beta_1: -1  Signifiance_Sum: 196  Power: 0.196"
## [1] "258  Sigma: 4  n: 10  beta_1: -0.9  Signifiance_Sum: 178  Power: 0.178"
## [1] "259  Sigma: 4  n: 10  beta_1: -0.8  Signifiance_Sum: 146  Power: 0.146"
## [1] "260  Sigma: 4  n: 10  beta_1: -0.7  Signifiance_Sum: 110  Power: 0.11"
## [1] "261  Sigma: 4  n: 10  beta_1: -0.6  Signifiance_Sum: 105  Power: 0.105"
## [1] "262  Sigma: 4  n: 10  beta_1: -0.5  Signifiance_Sum: 75  Power: 0.075"
## [1] "263  Sigma: 4  n: 10  beta_1: -0.4  Signifiance_Sum: 66  Power: 0.066"
## [1] "264  Sigma: 4  n: 10  beta_1: -0.3  Signifiance_Sum: 56  Power: 0.056"
## [1] "265  Sigma: 4  n: 10  beta_1: -0.2  Signifiance_Sum: 52  Power: 0.052"
## [1] "266  Sigma: 4  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 45  Power: 0.045"
## [1] "267  Sigma: 4  n: 10  beta_1: 0  Signifiance_Sum: 44  Power: 0.044"
## [1] "268  Sigma: 4  n: 10  beta_1: 0.1  Signifiance_Sum: 58  Power: 0.058"
## [1] "269  Sigma: 4  n: 10  beta_1: 0.2  Signifiance_Sum: 54  Power: 0.054"
## [1] "270  Sigma: 4  n: 10  beta_1: 0.3  Signifiance_Sum: 74  Power: 0.074"
## [1] "271  Sigma: 4  n: 10  beta_1: 0.4  Signifiance_Sum: 60  Power: 0.06"
## [1] "272  Sigma: 4  n: 10  beta_1: 0.5  Signifiance_Sum: 70  Power: 0.07"
## [1] "273  Sigma: 4  n: 10  beta_1: 0.6  Signifiance_Sum: 107  Power: 0.107"
## [1] "274  Sigma: 4  n: 10  beta_1: 0.7  Signifiance_Sum: 107  Power: 0.107"
## [1] "275  Sigma: 4  n: 10  beta_1: 0.8  Signifiance_Sum: 141  Power: 0.141"
## [1] "276  Sigma: 4  n: 10  beta_1: 0.9  Signifiance_Sum: 201  Power: 0.201"
## [1] "277  Sigma: 4  n: 10  beta_1: 1  Signifiance_Sum: 190  Power: 0.19"
## [1] "278  Sigma: 4  n: 10  beta_1: 1.1  Signifiance_Sum: 248  Power: 0.248"
## [1] "279  Sigma: 4  n: 10  beta_1: 1.2  Signifiance_Sum: 258  Power: 0.258"
## [1] "280  Sigma: 4  n: 10  beta_1: 1.3  Signifiance_Sum: 314  Power: 0.314"
## [1] "281  Sigma: 4  n: 10  beta_1: 1.4  Signifiance_Sum: 354  Power: 0.354"
## [1] "282  Sigma: 4  n: 10  beta_1: 1.5  Signifiance_Sum: 381  Power: 0.381"
## [1] "283  Sigma: 4  n: 10  beta_1: 1.6  Signifiance_Sum: 453  Power: 0.453"
## [1] "284  Sigma: 4  n: 10  beta_1: 1.7  Signifiance_Sum: 464  Power: 0.464"
## [1] "285  Sigma: 4  n: 10  beta_1: 1.8  Signifiance_Sum: 504  Power: 0.504"
## [1] "286  Sigma: 4  n: 10  beta_1: 1.9  Signifiance_Sum: 545  Power: 0.545"
## [1] "287  Sigma: 4  n: 10  beta_1: 2  Signifiance_Sum: 591  Power: 0.591"
## [1] "288  Sigma: 4  n: 20  beta_1: -2  Signifiance_Sum: 889  Power: 0.889"
## [1] "289  Sigma: 4  n: 20  beta_1: -1.9  Signifiance_Sum: 873  Power: 0.873"
## [1] "290  Sigma: 4  n: 20  beta_1: -1.8  Signifiance_Sum: 827  Power: 0.827"
## [1] "291  Sigma: 4  n: 20  beta_1: -1.7  Signifiance_Sum: 773  Power: 0.773"
## [1] "292  Sigma: 4  n: 20  beta_1: -1.6  Signifiance_Sum: 726  Power: 0.726"
## [1] "293  Sigma: 4  n: 20  beta_1: -1.5  Signifiance_Sum: 671  Power: 0.671"
## [1] "294  Sigma: 4  n: 20  beta_1: -1.4  Signifiance_Sum: 618  Power: 0.618"
## [1] "295  Sigma: 4  n: 20  beta_1: -1.3  Signifiance_Sum: 553  Power: 0.553"
## [1] "296  Sigma: 4  n: 20  beta_1: -1.2  Signifiance_Sum: 480  Power: 0.48"
## [1] "297  Sigma: 4  n: 20  beta_1: -1.1  Signifiance_Sum: 364  Power: 0.364"
## [1] "298  Sigma: 4  n: 20  beta_1: -1  Signifiance_Sum: 347  Power: 0.347"
## [1] "299  Sigma: 4  n: 20  beta_1: -0.9  Signifiance_Sum: 301  Power: 0.301"
## [1] "300  Sigma: 4  n: 20  beta_1: -0.8  Signifiance_Sum: 248  Power: 0.248"
## [1] "301  Sigma: 4  n: 20  beta_1: -0.7  Signifiance_Sum: 195  Power: 0.195"
## [1] "302  Sigma: 4  n: 20  beta_1: -0.6  Signifiance_Sum: 155  Power: 0.155"
## [1] "303  Sigma: 4  n: 20  beta_1: -0.5  Signifiance_Sum: 118  Power: 0.118"
## [1] "304  Sigma: 4  n: 20  beta_1: -0.4  Signifiance_Sum: 111  Power: 0.111"
## [1] "305  Sigma: 4  n: 20  beta_1: -0.3  Signifiance_Sum: 68  Power: 0.068"
## [1] "306  Sigma: 4  n: 20  beta_1: -0.2  Signifiance_Sum: 53  Power: 0.053"
## [1] "307  Sigma: 4  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 46  Power: 0.046"
## [1] "308  Sigma: 4  n: 20  beta_1: 0  Signifiance_Sum: 40  Power: 0.04"
## [1] "309  Sigma: 4  n: 20  beta_1: 0.1  Signifiance_Sum: 47  Power: 0.047"
## [1] "310  Sigma: 4  n: 20  beta_1: 0.2  Signifiance_Sum: 64  Power: 0.064"
## [1] "311  Sigma: 4  n: 20  beta_1: 0.3  Signifiance_Sum: 73  Power: 0.073"
## [1] "312  Sigma: 4  n: 20  beta_1: 0.4  Signifiance_Sum: 100  Power: 0.1"
## [1] "313  Sigma: 4  n: 20  beta_1: 0.5  Signifiance_Sum: 123  Power: 0.123"
## [1] "314  Sigma: 4  n: 20  beta_1: 0.6  Signifiance_Sum: 144  Power: 0.144"
## [1] "315  Sigma: 4  n: 20  beta_1: 0.7  Signifiance_Sum: 194  Power: 0.194"
## [1] "316  Sigma: 4  n: 20  beta_1: 0.8  Signifiance_Sum: 235  Power: 0.235"
## [1] "317  Sigma: 4  n: 20  beta_1: 0.9  Signifiance_Sum: 334  Power: 0.334"
## [1] "318  Sigma: 4  n: 20  beta_1: 1  Signifiance_Sum: 382  Power: 0.382"
## [1] "319  Sigma: 4  n: 20  beta_1: 1.1  Signifiance_Sum: 410  Power: 0.41"
## [1] "320  Sigma: 4  n: 20  beta_1: 1.2  Signifiance_Sum: 461  Power: 0.461"
## [1] "321  Sigma: 4  n: 20  beta_1: 1.3  Signifiance_Sum: 523  Power: 0.523"
## [1] "322  Sigma: 4  n: 20  beta_1: 1.4  Signifiance_Sum: 614  Power: 0.614"
## [1] "323  Sigma: 4  n: 20  beta_1: 1.5  Signifiance_Sum: 669  Power: 0.669"
## [1] "324  Sigma: 4  n: 20  beta_1: 1.6  Signifiance_Sum: 724  Power: 0.724"
## [1] "325  Sigma: 4  n: 20  beta_1: 1.7  Signifiance_Sum: 788  Power: 0.788"
## [1] "326  Sigma: 4  n: 20  beta_1: 1.8  Signifiance_Sum: 841  Power: 0.841"
## [1] "327  Sigma: 4  n: 20  beta_1: 1.9  Signifiance_Sum: 852  Power: 0.852"
## [1] "328  Sigma: 4  n: 20  beta_1: 2  Signifiance_Sum: 910  Power: 0.91"
## [1] "329  Sigma: 4  n: 30  beta_1: -2  Signifiance_Sum: 978  Power: 0.978"
## [1] "330  Sigma: 4  n: 30  beta_1: -1.9  Signifiance_Sum: 964  Power: 0.964"
## [1] "331  Sigma: 4  n: 30  beta_1: -1.8  Signifiance_Sum: 945  Power: 0.945"
## [1] "332  Sigma: 4  n: 30  beta_1: -1.7  Signifiance_Sum: 923  Power: 0.923"
## [1] "333  Sigma: 4  n: 30  beta_1: -1.6  Signifiance_Sum: 855  Power: 0.855"
## [1] "334  Sigma: 4  n: 30  beta_1: -1.5  Signifiance_Sum: 851  Power: 0.851"
## [1] "335  Sigma: 4  n: 30  beta_1: -1.4  Signifiance_Sum: 802  Power: 0.802"
## [1] "336  Sigma: 4  n: 30  beta_1: -1.3  Signifiance_Sum: 731  Power: 0.731"
## [1] "337  Sigma: 4  n: 30  beta_1: -1.2  Signifiance_Sum: 649  Power: 0.649"
## [1] "338  Sigma: 4  n: 30  beta_1: -1.1  Signifiance_Sum: 565  Power: 0.565"
## [1] "339  Sigma: 4  n: 30  beta_1: -1  Signifiance_Sum: 510  Power: 0.51"
## [1] "340  Sigma: 4  n: 30  beta_1: -0.9  Signifiance_Sum: 453  Power: 0.453"
## [1] "341  Sigma: 4  n: 30  beta_1: -0.8  Signifiance_Sum: 352  Power: 0.352"
## [1] "342  Sigma: 4  n: 30  beta_1: -0.7  Signifiance_Sum: 282  Power: 0.282"
## [1] "343  Sigma: 4  n: 30  beta_1: -0.6  Signifiance_Sum: 222  Power: 0.222"
## [1] "344  Sigma: 4  n: 30  beta_1: -0.5  Signifiance_Sum: 178  Power: 0.178"
## [1] "345  Sigma: 4  n: 30  beta_1: -0.4  Signifiance_Sum: 123  Power: 0.123"
## [1] "346  Sigma: 4  n: 30  beta_1: -0.3  Signifiance_Sum: 87  Power: 0.087"
## [1] "347  Sigma: 4  n: 30  beta_1: -0.2  Signifiance_Sum: 65  Power: 0.065"
## [1] "348  Sigma: 4  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 56  Power: 0.056"
## [1] "349  Sigma: 4  n: 30  beta_1: 0  Signifiance_Sum: 56  Power: 0.056"
## [1] "350  Sigma: 4  n: 30  beta_1: 0.1  Signifiance_Sum: 52  Power: 0.052"
## [1] "351  Sigma: 4  n: 30  beta_1: 0.2  Signifiance_Sum: 67  Power: 0.067"
## [1] "352  Sigma: 4  n: 30  beta_1: 0.3  Signifiance_Sum: 95  Power: 0.095"
## [1] "353  Sigma: 4  n: 30  beta_1: 0.4  Signifiance_Sum: 118  Power: 0.118"
## [1] "354  Sigma: 4  n: 30  beta_1: 0.5  Signifiance_Sum: 165  Power: 0.165"
## [1] "355  Sigma: 4  n: 30  beta_1: 0.6  Signifiance_Sum: 236  Power: 0.236"
## [1] "356  Sigma: 4  n: 30  beta_1: 0.7  Signifiance_Sum: 282  Power: 0.282"
## [1] "357  Sigma: 4  n: 30  beta_1: 0.8  Signifiance_Sum: 370  Power: 0.37"
## [1] "358  Sigma: 4  n: 30  beta_1: 0.9  Signifiance_Sum: 411  Power: 0.411"
## [1] "359  Sigma: 4  n: 30  beta_1: 1  Signifiance_Sum: 501  Power: 0.501"
## [1] "360  Sigma: 4  n: 30  beta_1: 1.1  Signifiance_Sum: 572  Power: 0.572"
## [1] "361  Sigma: 4  n: 30  beta_1: 1.2  Signifiance_Sum: 669  Power: 0.669"
## [1] "362  Sigma: 4  n: 30  beta_1: 1.3  Signifiance_Sum: 746  Power: 0.746"
## [1] "363  Sigma: 4  n: 30  beta_1: 1.4  Signifiance_Sum: 783  Power: 0.783"
## [1] "364  Sigma: 4  n: 30  beta_1: 1.5  Signifiance_Sum: 845  Power: 0.845"
## [1] "365  Sigma: 4  n: 30  beta_1: 1.6  Signifiance_Sum: 863  Power: 0.863"
## [1] "366  Sigma: 4  n: 30  beta_1: 1.7  Signifiance_Sum: 915  Power: 0.915"
## [1] "367  Sigma: 4  n: 30  beta_1: 1.8  Signifiance_Sum: 940  Power: 0.94"
## [1] "368  Sigma: 4  n: 30  beta_1: 1.9  Signifiance_Sum: 965  Power: 0.965"
## [1] "369  Sigma: 4  n: 30  beta_1: 2  Signifiance_Sum: 974  Power: 0.974"

[3.3] Results

As we have discussed in the Introduction on the aspect of Goal, we have to producded plots to support these Goals

  • Plot the Power Curve against the signal strength (β1) for different value of n, plot this kind of plot for each value of σ
#Below code plot the effect of Signal (β1) against the Power for noise of sigma = 1 against all sample size (10,20 and 30)
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 1",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,4], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))

The above plot depicts the following for sigma = 1

  • The power is 1 (i.e. # rejected hypothesis = # simulation) for beta_1 between -2 and -1 for all sample size (10,20 and 30)
  • As beta_1 increases after -1 and before 0, power decreases rapidly towards 0, means number of rejected hypothesis becomes small (failed to reject hypothesis becomes larger)
  • Again as beta_1 increases after 0, the power increases, i.e. number of rejected hypothesis is large.
  • Dropping of power nearly or after -1 depends on the sample size, more the sample size (30 > 20 > 10) power becomes 1 (i.e. # reject hypothesis is higher)
[3.3.1] Effect of Signal (β1) against Power
plot(model_sim[,1],model_sim[,5], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 2",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,6], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,6], col="red",lty=2)
points(model_sim[,1], model_sim[,7], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,7], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))

The above plot depicts the following for the sigma = 2

  • The power is 1 (i.e. # rejected hypothesis = # simulation) for beta_1 between till -1.5 for sample size 20 and 30. However lower the sample size with higher noise, the power decreases rapidly (i.e., # of hypothesis rejectes is small)
  • Once beta_1 increase after 0, power increases very rapidly for higher sample size, however slow for smaller sample size
  • As noise level increases (Sigma =2), the rate at which the power increases/ decreases also lower. So if we compare the power rate increases / decreases between sigma = 1 versus sigma = 2, it founds that power rate increase/decreases is much faster rate in sigma = 1 than that of sigma = 2
  • The distance between the power increase/decrease for different sample size (n=10/20 and 30) is wider for higher noise level (sigma = 2) than that of smaller noise level(sigma =1), which means reject the hypothesis is higher for higher sigma and even higher for lower sample size
[3.3.2] Below code plot the effect of Signal (β1) against the Power for noise of sigma = 4 against all sample size (10,20 and 30)
plot(model_sim[,1],model_sim[,8], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 4",xlab = "beta_1",ylab = "Power",ylim=c(0,1))
points(model_sim[,1], model_sim[,9], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,9], col="red",lty=2)
points(model_sim[,1], model_sim[,10], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,10], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))

The above plot depicts the following for tsigma = 4

  • The power seems lower at the initial stage of beta_1 itself, for the lower sample size (10) and decrease very fast until 0. However though it decreases very fast for higher sample size(n=20, 30), however relatively slow as compared to lower sample size (n=10)
  • The above point still valid for the case of increase of beta_1 beyond point 0, where power increases rapidly for higher sample size (30 and then 30) as compared to lower sample size
  • The distance between the power increase/decrease for different sample size (n=10/20 and 30) is wider for higher noise level (sigma = 4) than that of sigma =1 & 2, means rejecte the null hypothesis is higher for higher simga and even higher for lower sample size

[3.4] Discussion

Now as part of the goal, let’s discuss the what’s the impact of β1, n and σ with respect to power. We will be plotting different plots to conclude the summary of the impact

[3.4.1] Effect of sigma(1,2,4) on the Power **

plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Effect of sigma (1,2,4) on the Power",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="dodgerblue", pch=1,lty=1)
lines(model_sim[,1], model_sim[,3], col="dodgerblue",lty=1)
points(model_sim[,1], model_sim[,4], col="dodgerblue", pch=1,lty=1)
lines(model_sim[,1], model_sim[,4], col="dodgerblue",lty=1)

points(model_sim[,1],model_sim[,5], type="o", col="green", pch=1, lty=2)
lines(model_sim[,1],model_sim[,5], type="o", col="green", pch=1, lty=2)
points(model_sim[,1], model_sim[,6], col="green", pch=1,lty=2)
lines(model_sim[,1], model_sim[,6], col="green",lty=2)
points(model_sim[,1], model_sim[,7], col="green", pch=1,lty=2)
lines(model_sim[,1], model_sim[,7], col="green",lty=2)

points(model_sim[,1],model_sim[,8], type="o", col="red", pch=1, lty=3)
lines(model_sim[,1],model_sim[,8], type="o", col="red", pch=1, lty=3)
points(model_sim[,1], model_sim[,9], col="red", pch=1,lty=3)
lines(model_sim[,1], model_sim[,9], col="red",lty=3)
points(model_sim[,1], model_sim[,10], col="red", pch=1,lty=3)
lines(model_sim[,1], model_sim[,10], col="red",lty=3)
legend("topright",legend=c("sigma=1,n=10,20,30","sigma=2,n=10,20,30","sigma=3,n=10,20,30"),col=c("dodgerblue","green","darkorchid","red"),pch=c(1,1,1),lty=c(1,2,3))

Below are the observations on the effect of σ on power

  • Higher the sigma (σ),lower is the Power
  • It clearly visible that all the blue line (sigma=1) have higher power than that of green line (sigma = 2) & red line (sigma = 4) for any particular value of β1 and sample size. That means higher the value of the sigma (σ) = number of reject the hypothesis will be lower and lower the value of sigma (σ) = number of reject the null hypothesis becomes higher.
[3.4.2] Effect of n(10,20,30) on the Power **
par(mfrow=c(2, 2))

plot(model_sim[,1],model_sim[,2], type="o", col="deeppink", pch=1, lty=1,main="Sigma=1",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,3], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,4], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,4], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))

plot(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1,main="Sigma=2",xlab = "beta_1",ylab = "Power")
#points(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1)
lines(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1)
points(model_sim[,1], model_sim[,6], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,6], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,7], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,7], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))

plot(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1,main="Sigma=4",xlab = "beta_1",ylab = "Power",ylim=c(0,1))
#points(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1)
lines(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1)
points(model_sim[,1], model_sim[,9], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,9], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))

Below are the observations on the effect of sample size (n = 10,20 and 30) on the beta_1

  • From the abobe 3 plots, for different value of sigma (σ), Higher the sample size (n=30),higher is the power
  • It clearly visible that for all the 3 plots (against 3 different sigma (σ)), the blueviolet (n = 30) lines have higher than that of the rest 2 lines (green,n = 20 and deep pink, n= 10). That is higher the sample size, reject the hypothesis becomes higher
[3.4.3] Effect of n(10,20,30) on the Power **
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 1,2,4 & n = 10,20,30",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=2,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="tan1", pch=3,lty=3)
lines(model_sim[,1], model_sim[,4], col="tan1",lty=3)

points(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)

points(model_sim[,1], model_sim[,6], col="deeppink", pch=5,lty=5)
lines(model_sim[,1], model_sim[,6], col="deeppink",lty=5)
points(model_sim[,1], model_sim[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim[,1], model_sim[,7], col="darkolivegreen4",lty=6)

points(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)

points(model_sim[,1], model_sim[,9], col="darkblue", pch=8,lty=8)
lines(model_sim[,1], model_sim[,9], col="darkblue",lty=8)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=9)

legend("topright",legend=c("sigma=1,n=10","sigma=1,n=20","sigma=1,n=30","sigma=2,n=10","sigma=2,n=20","sigma=2,n=30","sigma=3,n=10","sigma=3,n=20","sigma=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))

If we combine all the plots for all the sample size beta(β1), sample size(n) and sigma(σ), from the above figure, below are the observations on the effect of β1 on power *

  • Power decreased as β1 increased between -2 and 0 for any given sigma(σ) and sample size(n), means as β1 increases, reject hypothesis decreases
  • Power increased as β1 increased between 0 and 2 for any given sigma(σ) and sample size(n), means as β1 increases, reject hypothesis increases

[3.4.4] Are 1000 simulations sufficient

Let’s simulate the MLR for 3000 iteration and record the Train / Test RMSE to see what’s the change

birthday = 19770411
set.seed(birthday)
no_loop = 3000

#Matrix to store the Power of the simulated models
model_sim_3000 = cbind(beta_1_sim_val=rep(0,length(beta_1)),n_10_sigma_1=rep(0,length(beta_1)),n_20_sigma_1=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)),n_10_sigma_2=rep(0,length(beta_1)),n_20_sigma_2=rep(0,length(beta_1)),n_30_sigma_2=rep(0,length(beta_1)),n_10_sigma_4=rep(0,length(beta_1)),n_20_sigma_4=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)))

model_sim_3000[,1] = seq(from=-2,to=2,by=0.1)
extra_counter = 0

#Loop to iterate over Sigma, number of samples, beta_1 values and 1000 simulations to extract and store the Power values
run_counter = 0
for(s in 1:length(sigma)){ #sigma = c(1,2,4)
  for (n_s in 1:length(n)){ #n = c(10,20,30)
    for (b in 1:length(beta_1)){ # beta_1 = seq(from=-2,to=2,by=0.1)
      tot_sum_signifincace_sim = 0
      run_counter = run_counter + 1      
      for(i in 1:no_loop){
        slr_data = slr_sim(n_in = n[n_s], sd = sigma[s], beta_1_in = beta_1[b] )
        model = lm(response~predictor, data = slr_data)
        model_p_val = summary(model)$coefficient[2,4]
        tot_sum_signifincace_sim = tot_sum_signifincace_sim + ifelse(model_p_val<alpha,1,0)
      }
      power_sim_beta = tot_sum_signifincace_sim / no_loop
      model_sim_3000[b,s+n_s+extra_counter] = power_sim_beta    
      print (paste(run_counter," Sigma:",sigma[s]," n:",n[n_s]," beta_1:",beta_1[b]," Signifiance_Sum:",tot_sum_signifincace_sim, " Power:",power_sim_beta))
    }
  }
  extra_counter = extra_counter + 2
}
## [1] "1  Sigma: 1  n: 10  beta_1: -2  Signifiance_Sum: 3000  Power: 1"
## [1] "2  Sigma: 1  n: 10  beta_1: -1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "3  Sigma: 1  n: 10  beta_1: -1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "4  Sigma: 1  n: 10  beta_1: -1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "5  Sigma: 1  n: 10  beta_1: -1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "6  Sigma: 1  n: 10  beta_1: -1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "7  Sigma: 1  n: 10  beta_1: -1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "8  Sigma: 1  n: 10  beta_1: -1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "9  Sigma: 1  n: 10  beta_1: -1.2  Signifiance_Sum: 2997  Power: 0.999"
## [1] "10  Sigma: 1  n: 10  beta_1: -1.1  Signifiance_Sum: 2993  Power: 0.997666666666667"
## [1] "11  Sigma: 1  n: 10  beta_1: -1  Signifiance_Sum: 2978  Power: 0.992666666666667"
## [1] "12  Sigma: 1  n: 10  beta_1: -0.9  Signifiance_Sum: 2938  Power: 0.979333333333333"
## [1] "13  Sigma: 1  n: 10  beta_1: -0.8  Signifiance_Sum: 2821  Power: 0.940333333333333"
## [1] "14  Sigma: 1  n: 10  beta_1: -0.7  Signifiance_Sum: 2610  Power: 0.87"
## [1] "15  Sigma: 1  n: 10  beta_1: -0.6  Signifiance_Sum: 2287  Power: 0.762333333333333"
## [1] "16  Sigma: 1  n: 10  beta_1: -0.5  Signifiance_Sum: 1839  Power: 0.613"
## [1] "17  Sigma: 1  n: 10  beta_1: -0.4  Signifiance_Sum: 1277  Power: 0.425666666666667"
## [1] "18  Sigma: 1  n: 10  beta_1: -0.3  Signifiance_Sum: 765  Power: 0.255"
## [1] "19  Sigma: 1  n: 10  beta_1: -0.2  Signifiance_Sum: 438  Power: 0.146"
## [1] "20  Sigma: 1  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 231  Power: 0.077"
## [1] "21  Sigma: 1  n: 10  beta_1: 0  Signifiance_Sum: 160  Power: 0.0533333333333333"
## [1] "22  Sigma: 1  n: 10  beta_1: 0.1  Signifiance_Sum: 227  Power: 0.0756666666666667"
## [1] "23  Sigma: 1  n: 10  beta_1: 0.2  Signifiance_Sum: 468  Power: 0.156"
## [1] "24  Sigma: 1  n: 10  beta_1: 0.3  Signifiance_Sum: 812  Power: 0.270666666666667"
## [1] "25  Sigma: 1  n: 10  beta_1: 0.4  Signifiance_Sum: 1277  Power: 0.425666666666667"
## [1] "26  Sigma: 1  n: 10  beta_1: 0.5  Signifiance_Sum: 1829  Power: 0.609666666666667"
## [1] "27  Sigma: 1  n: 10  beta_1: 0.6  Signifiance_Sum: 2239  Power: 0.746333333333333"
## [1] "28  Sigma: 1  n: 10  beta_1: 0.7  Signifiance_Sum: 2638  Power: 0.879333333333333"
## [1] "29  Sigma: 1  n: 10  beta_1: 0.8  Signifiance_Sum: 2798  Power: 0.932666666666667"
## [1] "30  Sigma: 1  n: 10  beta_1: 0.9  Signifiance_Sum: 2934  Power: 0.978"
## [1] "31  Sigma: 1  n: 10  beta_1: 1  Signifiance_Sum: 2981  Power: 0.993666666666667"
## [1] "32  Sigma: 1  n: 10  beta_1: 1.1  Signifiance_Sum: 2993  Power: 0.997666666666667"
## [1] "33  Sigma: 1  n: 10  beta_1: 1.2  Signifiance_Sum: 2997  Power: 0.999"
## [1] "34  Sigma: 1  n: 10  beta_1: 1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "35  Sigma: 1  n: 10  beta_1: 1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "36  Sigma: 1  n: 10  beta_1: 1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "37  Sigma: 1  n: 10  beta_1: 1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "38  Sigma: 1  n: 10  beta_1: 1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "39  Sigma: 1  n: 10  beta_1: 1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "40  Sigma: 1  n: 10  beta_1: 1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "41  Sigma: 1  n: 10  beta_1: 2  Signifiance_Sum: 3000  Power: 1"
## [1] "42  Sigma: 1  n: 20  beta_1: -2  Signifiance_Sum: 3000  Power: 1"
## [1] "43  Sigma: 1  n: 20  beta_1: -1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "44  Sigma: 1  n: 20  beta_1: -1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "45  Sigma: 1  n: 20  beta_1: -1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "46  Sigma: 1  n: 20  beta_1: -1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "47  Sigma: 1  n: 20  beta_1: -1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "48  Sigma: 1  n: 20  beta_1: -1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "49  Sigma: 1  n: 20  beta_1: -1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "50  Sigma: 1  n: 20  beta_1: -1.2  Signifiance_Sum: 3000  Power: 1"
## [1] "51  Sigma: 1  n: 20  beta_1: -1.1  Signifiance_Sum: 3000  Power: 1"
## [1] "52  Sigma: 1  n: 20  beta_1: -1  Signifiance_Sum: 3000  Power: 1"
## [1] "53  Sigma: 1  n: 20  beta_1: -0.9  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "54  Sigma: 1  n: 20  beta_1: -0.8  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "55  Sigma: 1  n: 20  beta_1: -0.7  Signifiance_Sum: 2986  Power: 0.995333333333333"
## [1] "56  Sigma: 1  n: 20  beta_1: -0.6  Signifiance_Sum: 2933  Power: 0.977666666666667"
## [1] "57  Sigma: 1  n: 20  beta_1: -0.5  Signifiance_Sum: 2663  Power: 0.887666666666667"
## [1] "58  Sigma: 1  n: 20  beta_1: -0.4  Signifiance_Sum: 2199  Power: 0.733"
## [1] "59  Sigma: 1  n: 20  beta_1: -0.3  Signifiance_Sum: 1498  Power: 0.499333333333333"
## [1] "60  Sigma: 1  n: 20  beta_1: -0.2  Signifiance_Sum: 730  Power: 0.243333333333333"
## [1] "61  Sigma: 1  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 303  Power: 0.101"
## [1] "62  Sigma: 1  n: 20  beta_1: 0  Signifiance_Sum: 157  Power: 0.0523333333333333"
## [1] "63  Sigma: 1  n: 20  beta_1: 0.1  Signifiance_Sum: 301  Power: 0.100333333333333"
## [1] "64  Sigma: 1  n: 20  beta_1: 0.2  Signifiance_Sum: 704  Power: 0.234666666666667"
## [1] "65  Sigma: 1  n: 20  beta_1: 0.3  Signifiance_Sum: 1458  Power: 0.486"
## [1] "66  Sigma: 1  n: 20  beta_1: 0.4  Signifiance_Sum: 2182  Power: 0.727333333333333"
## [1] "67  Sigma: 1  n: 20  beta_1: 0.5  Signifiance_Sum: 2648  Power: 0.882666666666667"
## [1] "68  Sigma: 1  n: 20  beta_1: 0.6  Signifiance_Sum: 2915  Power: 0.971666666666667"
## [1] "69  Sigma: 1  n: 20  beta_1: 0.7  Signifiance_Sum: 2984  Power: 0.994666666666667"
## [1] "70  Sigma: 1  n: 20  beta_1: 0.8  Signifiance_Sum: 2998  Power: 0.999333333333333"
## [1] "71  Sigma: 1  n: 20  beta_1: 0.9  Signifiance_Sum: 3000  Power: 1"
## [1] "72  Sigma: 1  n: 20  beta_1: 1  Signifiance_Sum: 3000  Power: 1"
## [1] "73  Sigma: 1  n: 20  beta_1: 1.1  Signifiance_Sum: 3000  Power: 1"
## [1] "74  Sigma: 1  n: 20  beta_1: 1.2  Signifiance_Sum: 3000  Power: 1"
## [1] "75  Sigma: 1  n: 20  beta_1: 1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "76  Sigma: 1  n: 20  beta_1: 1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "77  Sigma: 1  n: 20  beta_1: 1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "78  Sigma: 1  n: 20  beta_1: 1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "79  Sigma: 1  n: 20  beta_1: 1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "80  Sigma: 1  n: 20  beta_1: 1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "81  Sigma: 1  n: 20  beta_1: 1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "82  Sigma: 1  n: 20  beta_1: 2  Signifiance_Sum: 3000  Power: 1"
## [1] "83  Sigma: 1  n: 30  beta_1: -2  Signifiance_Sum: 3000  Power: 1"
## [1] "84  Sigma: 1  n: 30  beta_1: -1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "85  Sigma: 1  n: 30  beta_1: -1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "86  Sigma: 1  n: 30  beta_1: -1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "87  Sigma: 1  n: 30  beta_1: -1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "88  Sigma: 1  n: 30  beta_1: -1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "89  Sigma: 1  n: 30  beta_1: -1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "90  Sigma: 1  n: 30  beta_1: -1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "91  Sigma: 1  n: 30  beta_1: -1.2  Signifiance_Sum: 3000  Power: 1"
## [1] "92  Sigma: 1  n: 30  beta_1: -1.1  Signifiance_Sum: 3000  Power: 1"
## [1] "93  Sigma: 1  n: 30  beta_1: -1  Signifiance_Sum: 3000  Power: 1"
## [1] "94  Sigma: 1  n: 30  beta_1: -0.9  Signifiance_Sum: 3000  Power: 1"
## [1] "95  Sigma: 1  n: 30  beta_1: -0.8  Signifiance_Sum: 3000  Power: 1"
## [1] "96  Sigma: 1  n: 30  beta_1: -0.7  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "97  Sigma: 1  n: 30  beta_1: -0.6  Signifiance_Sum: 2993  Power: 0.997666666666667"
## [1] "98  Sigma: 1  n: 30  beta_1: -0.5  Signifiance_Sum: 2916  Power: 0.972"
## [1] "99  Sigma: 1  n: 30  beta_1: -0.4  Signifiance_Sum: 2654  Power: 0.884666666666667"
## [1] "100  Sigma: 1  n: 30  beta_1: -0.3  Signifiance_Sum: 1949  Power: 0.649666666666667"
## [1] "101  Sigma: 1  n: 30  beta_1: -0.2  Signifiance_Sum: 1007  Power: 0.335666666666667"
## [1] "102  Sigma: 1  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 374  Power: 0.124666666666667"
## [1] "103  Sigma: 1  n: 30  beta_1: 0  Signifiance_Sum: 152  Power: 0.0506666666666667"
## [1] "104  Sigma: 1  n: 30  beta_1: 0.1  Signifiance_Sum: 344  Power: 0.114666666666667"
## [1] "105  Sigma: 1  n: 30  beta_1: 0.2  Signifiance_Sum: 1078  Power: 0.359333333333333"
## [1] "106  Sigma: 1  n: 30  beta_1: 0.3  Signifiance_Sum: 1940  Power: 0.646666666666667"
## [1] "107  Sigma: 1  n: 30  beta_1: 0.4  Signifiance_Sum: 2683  Power: 0.894333333333333"
## [1] "108  Sigma: 1  n: 30  beta_1: 0.5  Signifiance_Sum: 2939  Power: 0.979666666666667"
## [1] "109  Sigma: 1  n: 30  beta_1: 0.6  Signifiance_Sum: 2987  Power: 0.995666666666667"
## [1] "110  Sigma: 1  n: 30  beta_1: 0.7  Signifiance_Sum: 3000  Power: 1"
## [1] "111  Sigma: 1  n: 30  beta_1: 0.8  Signifiance_Sum: 3000  Power: 1"
## [1] "112  Sigma: 1  n: 30  beta_1: 0.9  Signifiance_Sum: 3000  Power: 1"
## [1] "113  Sigma: 1  n: 30  beta_1: 1  Signifiance_Sum: 3000  Power: 1"
## [1] "114  Sigma: 1  n: 30  beta_1: 1.1  Signifiance_Sum: 3000  Power: 1"
## [1] "115  Sigma: 1  n: 30  beta_1: 1.2  Signifiance_Sum: 3000  Power: 1"
## [1] "116  Sigma: 1  n: 30  beta_1: 1.3  Signifiance_Sum: 3000  Power: 1"
## [1] "117  Sigma: 1  n: 30  beta_1: 1.4  Signifiance_Sum: 3000  Power: 1"
## [1] "118  Sigma: 1  n: 30  beta_1: 1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "119  Sigma: 1  n: 30  beta_1: 1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "120  Sigma: 1  n: 30  beta_1: 1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "121  Sigma: 1  n: 30  beta_1: 1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "122  Sigma: 1  n: 30  beta_1: 1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "123  Sigma: 1  n: 30  beta_1: 2  Signifiance_Sum: 3000  Power: 1"
## [1] "124  Sigma: 2  n: 10  beta_1: -2  Signifiance_Sum: 2972  Power: 0.990666666666667"
## [1] "125  Sigma: 2  n: 10  beta_1: -1.9  Signifiance_Sum: 2955  Power: 0.985"
## [1] "126  Sigma: 2  n: 10  beta_1: -1.8  Signifiance_Sum: 2922  Power: 0.974"
## [1] "127  Sigma: 2  n: 10  beta_1: -1.7  Signifiance_Sum: 2878  Power: 0.959333333333333"
## [1] "128  Sigma: 2  n: 10  beta_1: -1.6  Signifiance_Sum: 2845  Power: 0.948333333333333"
## [1] "129  Sigma: 2  n: 10  beta_1: -1.5  Signifiance_Sum: 2737  Power: 0.912333333333333"
## [1] "130  Sigma: 2  n: 10  beta_1: -1.4  Signifiance_Sum: 2614  Power: 0.871333333333333"
## [1] "131  Sigma: 2  n: 10  beta_1: -1.3  Signifiance_Sum: 2457  Power: 0.819"
## [1] "132  Sigma: 2  n: 10  beta_1: -1.2  Signifiance_Sum: 2296  Power: 0.765333333333333"
## [1] "133  Sigma: 2  n: 10  beta_1: -1.1  Signifiance_Sum: 2027  Power: 0.675666666666667"
## [1] "134  Sigma: 2  n: 10  beta_1: -1  Signifiance_Sum: 1818  Power: 0.606"
## [1] "135  Sigma: 2  n: 10  beta_1: -0.9  Signifiance_Sum: 1583  Power: 0.527666666666667"
## [1] "136  Sigma: 2  n: 10  beta_1: -0.8  Signifiance_Sum: 1299  Power: 0.433"
## [1] "137  Sigma: 2  n: 10  beta_1: -0.7  Signifiance_Sum: 1031  Power: 0.343666666666667"
## [1] "138  Sigma: 2  n: 10  beta_1: -0.6  Signifiance_Sum: 811  Power: 0.270333333333333"
## [1] "139  Sigma: 2  n: 10  beta_1: -0.5  Signifiance_Sum: 601  Power: 0.200333333333333"
## [1] "140  Sigma: 2  n: 10  beta_1: -0.4  Signifiance_Sum: 420  Power: 0.14"
## [1] "141  Sigma: 2  n: 10  beta_1: -0.3  Signifiance_Sum: 300  Power: 0.1"
## [1] "142  Sigma: 2  n: 10  beta_1: -0.2  Signifiance_Sum: 209  Power: 0.0696666666666667"
## [1] "143  Sigma: 2  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 156  Power: 0.052"
## [1] "144  Sigma: 2  n: 10  beta_1: 0  Signifiance_Sum: 154  Power: 0.0513333333333333"
## [1] "145  Sigma: 2  n: 10  beta_1: 0.1  Signifiance_Sum: 180  Power: 0.06"
## [1] "146  Sigma: 2  n: 10  beta_1: 0.2  Signifiance_Sum: 209  Power: 0.0696666666666667"
## [1] "147  Sigma: 2  n: 10  beta_1: 0.3  Signifiance_Sum: 332  Power: 0.110666666666667"
## [1] "148  Sigma: 2  n: 10  beta_1: 0.4  Signifiance_Sum: 471  Power: 0.157"
## [1] "149  Sigma: 2  n: 10  beta_1: 0.5  Signifiance_Sum: 577  Power: 0.192333333333333"
## [1] "150  Sigma: 2  n: 10  beta_1: 0.6  Signifiance_Sum: 809  Power: 0.269666666666667"
## [1] "151  Sigma: 2  n: 10  beta_1: 0.7  Signifiance_Sum: 1046  Power: 0.348666666666667"
## [1] "152  Sigma: 2  n: 10  beta_1: 0.8  Signifiance_Sum: 1280  Power: 0.426666666666667"
## [1] "153  Sigma: 2  n: 10  beta_1: 0.9  Signifiance_Sum: 1534  Power: 0.511333333333333"
## [1] "154  Sigma: 2  n: 10  beta_1: 1  Signifiance_Sum: 1773  Power: 0.591"
## [1] "155  Sigma: 2  n: 10  beta_1: 1.1  Signifiance_Sum: 2078  Power: 0.692666666666667"
## [1] "156  Sigma: 2  n: 10  beta_1: 1.2  Signifiance_Sum: 2266  Power: 0.755333333333333"
## [1] "157  Sigma: 2  n: 10  beta_1: 1.3  Signifiance_Sum: 2445  Power: 0.815"
## [1] "158  Sigma: 2  n: 10  beta_1: 1.4  Signifiance_Sum: 2652  Power: 0.884"
## [1] "159  Sigma: 2  n: 10  beta_1: 1.5  Signifiance_Sum: 2733  Power: 0.911"
## [1] "160  Sigma: 2  n: 10  beta_1: 1.6  Signifiance_Sum: 2823  Power: 0.941"
## [1] "161  Sigma: 2  n: 10  beta_1: 1.7  Signifiance_Sum: 2895  Power: 0.965"
## [1] "162  Sigma: 2  n: 10  beta_1: 1.8  Signifiance_Sum: 2931  Power: 0.977"
## [1] "163  Sigma: 2  n: 10  beta_1: 1.9  Signifiance_Sum: 2961  Power: 0.987"
## [1] "164  Sigma: 2  n: 10  beta_1: 2  Signifiance_Sum: 2978  Power: 0.992666666666667"
## [1] "165  Sigma: 2  n: 20  beta_1: -2  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "166  Sigma: 2  n: 20  beta_1: -1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "167  Sigma: 2  n: 20  beta_1: -1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "168  Sigma: 2  n: 20  beta_1: -1.7  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "169  Sigma: 2  n: 20  beta_1: -1.6  Signifiance_Sum: 2996  Power: 0.998666666666667"
## [1] "170  Sigma: 2  n: 20  beta_1: -1.5  Signifiance_Sum: 2986  Power: 0.995333333333333"
## [1] "171  Sigma: 2  n: 20  beta_1: -1.4  Signifiance_Sum: 2985  Power: 0.995"
## [1] "172  Sigma: 2  n: 20  beta_1: -1.3  Signifiance_Sum: 2974  Power: 0.991333333333333"
## [1] "173  Sigma: 2  n: 20  beta_1: -1.2  Signifiance_Sum: 2904  Power: 0.968"
## [1] "174  Sigma: 2  n: 20  beta_1: -1.1  Signifiance_Sum: 2824  Power: 0.941333333333333"
## [1] "175  Sigma: 2  n: 20  beta_1: -1  Signifiance_Sum: 2683  Power: 0.894333333333333"
## [1] "176  Sigma: 2  n: 20  beta_1: -0.9  Signifiance_Sum: 2487  Power: 0.829"
## [1] "177  Sigma: 2  n: 20  beta_1: -0.8  Signifiance_Sum: 2202  Power: 0.734"
## [1] "178  Sigma: 2  n: 20  beta_1: -0.7  Signifiance_Sum: 1840  Power: 0.613333333333333"
## [1] "179  Sigma: 2  n: 20  beta_1: -0.6  Signifiance_Sum: 1453  Power: 0.484333333333333"
## [1] "180  Sigma: 2  n: 20  beta_1: -0.5  Signifiance_Sum: 1088  Power: 0.362666666666667"
## [1] "181  Sigma: 2  n: 20  beta_1: -0.4  Signifiance_Sum: 756  Power: 0.252"
## [1] "182  Sigma: 2  n: 20  beta_1: -0.3  Signifiance_Sum: 531  Power: 0.177"
## [1] "183  Sigma: 2  n: 20  beta_1: -0.2  Signifiance_Sum: 305  Power: 0.101666666666667"
## [1] "184  Sigma: 2  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 184  Power: 0.0613333333333333"
## [1] "185  Sigma: 2  n: 20  beta_1: 0  Signifiance_Sum: 130  Power: 0.0433333333333333"
## [1] "186  Sigma: 2  n: 20  beta_1: 0.1  Signifiance_Sum: 202  Power: 0.0673333333333333"
## [1] "187  Sigma: 2  n: 20  beta_1: 0.2  Signifiance_Sum: 299  Power: 0.0996666666666667"
## [1] "188  Sigma: 2  n: 20  beta_1: 0.3  Signifiance_Sum: 482  Power: 0.160666666666667"
## [1] "189  Sigma: 2  n: 20  beta_1: 0.4  Signifiance_Sum: 746  Power: 0.248666666666667"
## [1] "190  Sigma: 2  n: 20  beta_1: 0.5  Signifiance_Sum: 1050  Power: 0.35"
## [1] "191  Sigma: 2  n: 20  beta_1: 0.6  Signifiance_Sum: 1457  Power: 0.485666666666667"
## [1] "192  Sigma: 2  n: 20  beta_1: 0.7  Signifiance_Sum: 1854  Power: 0.618"
## [1] "193  Sigma: 2  n: 20  beta_1: 0.8  Signifiance_Sum: 2156  Power: 0.718666666666667"
## [1] "194  Sigma: 2  n: 20  beta_1: 0.9  Signifiance_Sum: 2521  Power: 0.840333333333333"
## [1] "195  Sigma: 2  n: 20  beta_1: 1  Signifiance_Sum: 2704  Power: 0.901333333333333"
## [1] "196  Sigma: 2  n: 20  beta_1: 1.1  Signifiance_Sum: 2823  Power: 0.941"
## [1] "197  Sigma: 2  n: 20  beta_1: 1.2  Signifiance_Sum: 2912  Power: 0.970666666666667"
## [1] "198  Sigma: 2  n: 20  beta_1: 1.3  Signifiance_Sum: 2966  Power: 0.988666666666667"
## [1] "199  Sigma: 2  n: 20  beta_1: 1.4  Signifiance_Sum: 2990  Power: 0.996666666666667"
## [1] "200  Sigma: 2  n: 20  beta_1: 1.5  Signifiance_Sum: 2993  Power: 0.997666666666667"
## [1] "201  Sigma: 2  n: 20  beta_1: 1.6  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "202  Sigma: 2  n: 20  beta_1: 1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "203  Sigma: 2  n: 20  beta_1: 1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "204  Sigma: 2  n: 20  beta_1: 1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "205  Sigma: 2  n: 20  beta_1: 2  Signifiance_Sum: 3000  Power: 1"
## [1] "206  Sigma: 2  n: 30  beta_1: -2  Signifiance_Sum: 3000  Power: 1"
## [1] "207  Sigma: 2  n: 30  beta_1: -1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "208  Sigma: 2  n: 30  beta_1: -1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "209  Sigma: 2  n: 30  beta_1: -1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "210  Sigma: 2  n: 30  beta_1: -1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "211  Sigma: 2  n: 30  beta_1: -1.5  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "212  Sigma: 2  n: 30  beta_1: -1.4  Signifiance_Sum: 2999  Power: 0.999666666666667"
## [1] "213  Sigma: 2  n: 30  beta_1: -1.3  Signifiance_Sum: 2995  Power: 0.998333333333333"
## [1] "214  Sigma: 2  n: 30  beta_1: -1.2  Signifiance_Sum: 2991  Power: 0.997"
## [1] "215  Sigma: 2  n: 30  beta_1: -1.1  Signifiance_Sum: 2964  Power: 0.988"
## [1] "216  Sigma: 2  n: 30  beta_1: -1  Signifiance_Sum: 2933  Power: 0.977666666666667"
## [1] "217  Sigma: 2  n: 30  beta_1: -0.9  Signifiance_Sum: 2811  Power: 0.937"
## [1] "218  Sigma: 2  n: 30  beta_1: -0.8  Signifiance_Sum: 2653  Power: 0.884333333333333"
## [1] "219  Sigma: 2  n: 30  beta_1: -0.7  Signifiance_Sum: 2386  Power: 0.795333333333333"
## [1] "220  Sigma: 2  n: 30  beta_1: -0.6  Signifiance_Sum: 1967  Power: 0.655666666666667"
## [1] "221  Sigma: 2  n: 30  beta_1: -0.5  Signifiance_Sum: 1466  Power: 0.488666666666667"
## [1] "222  Sigma: 2  n: 30  beta_1: -0.4  Signifiance_Sum: 1050  Power: 0.35"
## [1] "223  Sigma: 2  n: 30  beta_1: -0.3  Signifiance_Sum: 632  Power: 0.210666666666667"
## [1] "224  Sigma: 2  n: 30  beta_1: -0.2  Signifiance_Sum: 385  Power: 0.128333333333333"
## [1] "225  Sigma: 2  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 194  Power: 0.0646666666666667"
## [1] "226  Sigma: 2  n: 30  beta_1: 0  Signifiance_Sum: 129  Power: 0.043"
## [1] "227  Sigma: 2  n: 30  beta_1: 0.1  Signifiance_Sum: 181  Power: 0.0603333333333333"
## [1] "228  Sigma: 2  n: 30  beta_1: 0.2  Signifiance_Sum: 388  Power: 0.129333333333333"
## [1] "229  Sigma: 2  n: 30  beta_1: 0.3  Signifiance_Sum: 650  Power: 0.216666666666667"
## [1] "230  Sigma: 2  n: 30  beta_1: 0.4  Signifiance_Sum: 1014  Power: 0.338"
## [1] "231  Sigma: 2  n: 30  beta_1: 0.5  Signifiance_Sum: 1490  Power: 0.496666666666667"
## [1] "232  Sigma: 2  n: 30  beta_1: 0.6  Signifiance_Sum: 1963  Power: 0.654333333333333"
## [1] "233  Sigma: 2  n: 30  beta_1: 0.7  Signifiance_Sum: 2371  Power: 0.790333333333333"
## [1] "234  Sigma: 2  n: 30  beta_1: 0.8  Signifiance_Sum: 2637  Power: 0.879"
## [1] "235  Sigma: 2  n: 30  beta_1: 0.9  Signifiance_Sum: 2857  Power: 0.952333333333333"
## [1] "236  Sigma: 2  n: 30  beta_1: 1  Signifiance_Sum: 2943  Power: 0.981"
## [1] "237  Sigma: 2  n: 30  beta_1: 1.1  Signifiance_Sum: 2971  Power: 0.990333333333333"
## [1] "238  Sigma: 2  n: 30  beta_1: 1.2  Signifiance_Sum: 2995  Power: 0.998333333333333"
## [1] "239  Sigma: 2  n: 30  beta_1: 1.3  Signifiance_Sum: 2998  Power: 0.999333333333333"
## [1] "240  Sigma: 2  n: 30  beta_1: 1.4  Signifiance_Sum: 2998  Power: 0.999333333333333"
## [1] "241  Sigma: 2  n: 30  beta_1: 1.5  Signifiance_Sum: 3000  Power: 1"
## [1] "242  Sigma: 2  n: 30  beta_1: 1.6  Signifiance_Sum: 3000  Power: 1"
## [1] "243  Sigma: 2  n: 30  beta_1: 1.7  Signifiance_Sum: 3000  Power: 1"
## [1] "244  Sigma: 2  n: 30  beta_1: 1.8  Signifiance_Sum: 3000  Power: 1"
## [1] "245  Sigma: 2  n: 30  beta_1: 1.9  Signifiance_Sum: 3000  Power: 1"
## [1] "246  Sigma: 2  n: 30  beta_1: 2  Signifiance_Sum: 3000  Power: 1"
## [1] "247  Sigma: 4  n: 10  beta_1: -2  Signifiance_Sum: 1790  Power: 0.596666666666667"
## [1] "248  Sigma: 4  n: 10  beta_1: -1.9  Signifiance_Sum: 1657  Power: 0.552333333333333"
## [1] "249  Sigma: 4  n: 10  beta_1: -1.8  Signifiance_Sum: 1543  Power: 0.514333333333333"
## [1] "250  Sigma: 4  n: 10  beta_1: -1.7  Signifiance_Sum: 1428  Power: 0.476"
## [1] "251  Sigma: 4  n: 10  beta_1: -1.6  Signifiance_Sum: 1273  Power: 0.424333333333333"
## [1] "252  Sigma: 4  n: 10  beta_1: -1.5  Signifiance_Sum: 1182  Power: 0.394"
## [1] "253  Sigma: 4  n: 10  beta_1: -1.4  Signifiance_Sum: 1026  Power: 0.342"
## [1] "254  Sigma: 4  n: 10  beta_1: -1.3  Signifiance_Sum: 919  Power: 0.306333333333333"
## [1] "255  Sigma: 4  n: 10  beta_1: -1.2  Signifiance_Sum: 798  Power: 0.266"
## [1] "256  Sigma: 4  n: 10  beta_1: -1.1  Signifiance_Sum: 692  Power: 0.230666666666667"
## [1] "257  Sigma: 4  n: 10  beta_1: -1  Signifiance_Sum: 580  Power: 0.193333333333333"
## [1] "258  Sigma: 4  n: 10  beta_1: -0.9  Signifiance_Sum: 517  Power: 0.172333333333333"
## [1] "259  Sigma: 4  n: 10  beta_1: -0.8  Signifiance_Sum: 443  Power: 0.147666666666667"
## [1] "260  Sigma: 4  n: 10  beta_1: -0.7  Signifiance_Sum: 364  Power: 0.121333333333333"
## [1] "261  Sigma: 4  n: 10  beta_1: -0.6  Signifiance_Sum: 323  Power: 0.107666666666667"
## [1] "262  Sigma: 4  n: 10  beta_1: -0.5  Signifiance_Sum: 285  Power: 0.095"
## [1] "263  Sigma: 4  n: 10  beta_1: -0.4  Signifiance_Sum: 247  Power: 0.0823333333333333"
## [1] "264  Sigma: 4  n: 10  beta_1: -0.3  Signifiance_Sum: 196  Power: 0.0653333333333333"
## [1] "265  Sigma: 4  n: 10  beta_1: -0.2  Signifiance_Sum: 155  Power: 0.0516666666666667"
## [1] "266  Sigma: 4  n: 10  beta_1: -0.0999999999999999  Signifiance_Sum: 162  Power: 0.054"
## [1] "267  Sigma: 4  n: 10  beta_1: 0  Signifiance_Sum: 149  Power: 0.0496666666666667"
## [1] "268  Sigma: 4  n: 10  beta_1: 0.1  Signifiance_Sum: 133  Power: 0.0443333333333333"
## [1] "269  Sigma: 4  n: 10  beta_1: 0.2  Signifiance_Sum: 176  Power: 0.0586666666666667"
## [1] "270  Sigma: 4  n: 10  beta_1: 0.3  Signifiance_Sum: 184  Power: 0.0613333333333333"
## [1] "271  Sigma: 4  n: 10  beta_1: 0.4  Signifiance_Sum: 233  Power: 0.0776666666666667"
## [1] "272  Sigma: 4  n: 10  beta_1: 0.5  Signifiance_Sum: 273  Power: 0.091"
## [1] "273  Sigma: 4  n: 10  beta_1: 0.6  Signifiance_Sum: 302  Power: 0.100666666666667"
## [1] "274  Sigma: 4  n: 10  beta_1: 0.7  Signifiance_Sum: 334  Power: 0.111333333333333"
## [1] "275  Sigma: 4  n: 10  beta_1: 0.8  Signifiance_Sum: 431  Power: 0.143666666666667"
## [1] "276  Sigma: 4  n: 10  beta_1: 0.9  Signifiance_Sum: 501  Power: 0.167"
## [1] "277  Sigma: 4  n: 10  beta_1: 1  Signifiance_Sum: 597  Power: 0.199"
## [1] "278  Sigma: 4  n: 10  beta_1: 1.1  Signifiance_Sum: 707  Power: 0.235666666666667"
## [1] "279  Sigma: 4  n: 10  beta_1: 1.2  Signifiance_Sum: 828  Power: 0.276"
## [1] "280  Sigma: 4  n: 10  beta_1: 1.3  Signifiance_Sum: 931  Power: 0.310333333333333"
## [1] "281  Sigma: 4  n: 10  beta_1: 1.4  Signifiance_Sum: 1028  Power: 0.342666666666667"
## [1] "282  Sigma: 4  n: 10  beta_1: 1.5  Signifiance_Sum: 1131  Power: 0.377"
## [1] "283  Sigma: 4  n: 10  beta_1: 1.6  Signifiance_Sum: 1264  Power: 0.421333333333333"
## [1] "284  Sigma: 4  n: 10  beta_1: 1.7  Signifiance_Sum: 1371  Power: 0.457"
## [1] "285  Sigma: 4  n: 10  beta_1: 1.8  Signifiance_Sum: 1531  Power: 0.510333333333333"
## [1] "286  Sigma: 4  n: 10  beta_1: 1.9  Signifiance_Sum: 1643  Power: 0.547666666666667"
## [1] "287  Sigma: 4  n: 10  beta_1: 2  Signifiance_Sum: 1820  Power: 0.606666666666667"
## [1] "288  Sigma: 4  n: 20  beta_1: -2  Signifiance_Sum: 2686  Power: 0.895333333333333"
## [1] "289  Sigma: 4  n: 20  beta_1: -1.9  Signifiance_Sum: 2611  Power: 0.870333333333333"
## [1] "290  Sigma: 4  n: 20  beta_1: -1.8  Signifiance_Sum: 2474  Power: 0.824666666666667"
## [1] "291  Sigma: 4  n: 20  beta_1: -1.7  Signifiance_Sum: 2358  Power: 0.786"
## [1] "292  Sigma: 4  n: 20  beta_1: -1.6  Signifiance_Sum: 2180  Power: 0.726666666666667"
## [1] "293  Sigma: 4  n: 20  beta_1: -1.5  Signifiance_Sum: 2012  Power: 0.670666666666667"
## [1] "294  Sigma: 4  n: 20  beta_1: -1.4  Signifiance_Sum: 1803  Power: 0.601"
## [1] "295  Sigma: 4  n: 20  beta_1: -1.3  Signifiance_Sum: 1662  Power: 0.554"
## [1] "296  Sigma: 4  n: 20  beta_1: -1.2  Signifiance_Sum: 1469  Power: 0.489666666666667"
## [1] "297  Sigma: 4  n: 20  beta_1: -1.1  Signifiance_Sum: 1311  Power: 0.437"
## [1] "298  Sigma: 4  n: 20  beta_1: -1  Signifiance_Sum: 1074  Power: 0.358"
## [1] "299  Sigma: 4  n: 20  beta_1: -0.9  Signifiance_Sum: 926  Power: 0.308666666666667"
## [1] "300  Sigma: 4  n: 20  beta_1: -0.8  Signifiance_Sum: 763  Power: 0.254333333333333"
## [1] "301  Sigma: 4  n: 20  beta_1: -0.7  Signifiance_Sum: 601  Power: 0.200333333333333"
## [1] "302  Sigma: 4  n: 20  beta_1: -0.6  Signifiance_Sum: 533  Power: 0.177666666666667"
## [1] "303  Sigma: 4  n: 20  beta_1: -0.5  Signifiance_Sum: 383  Power: 0.127666666666667"
## [1] "304  Sigma: 4  n: 20  beta_1: -0.4  Signifiance_Sum: 303  Power: 0.101"
## [1] "305  Sigma: 4  n: 20  beta_1: -0.3  Signifiance_Sum: 251  Power: 0.0836666666666667"
## [1] "306  Sigma: 4  n: 20  beta_1: -0.2  Signifiance_Sum: 164  Power: 0.0546666666666667"
## [1] "307  Sigma: 4  n: 20  beta_1: -0.0999999999999999  Signifiance_Sum: 194  Power: 0.0646666666666667"
## [1] "308  Sigma: 4  n: 20  beta_1: 0  Signifiance_Sum: 133  Power: 0.0443333333333333"
## [1] "309  Sigma: 4  n: 20  beta_1: 0.1  Signifiance_Sum: 159  Power: 0.053"
## [1] "310  Sigma: 4  n: 20  beta_1: 0.2  Signifiance_Sum: 198  Power: 0.066"
## [1] "311  Sigma: 4  n: 20  beta_1: 0.3  Signifiance_Sum: 219  Power: 0.073"
## [1] "312  Sigma: 4  n: 20  beta_1: 0.4  Signifiance_Sum: 309  Power: 0.103"
## [1] "313  Sigma: 4  n: 20  beta_1: 0.5  Signifiance_Sum: 360  Power: 0.12"
## [1] "314  Sigma: 4  n: 20  beta_1: 0.6  Signifiance_Sum: 482  Power: 0.160666666666667"
## [1] "315  Sigma: 4  n: 20  beta_1: 0.7  Signifiance_Sum: 591  Power: 0.197"
## [1] "316  Sigma: 4  n: 20  beta_1: 0.8  Signifiance_Sum: 729  Power: 0.243"
## [1] "317  Sigma: 4  n: 20  beta_1: 0.9  Signifiance_Sum: 916  Power: 0.305333333333333"
## [1] "318  Sigma: 4  n: 20  beta_1: 1  Signifiance_Sum: 1120  Power: 0.373333333333333"
## [1] "319  Sigma: 4  n: 20  beta_1: 1.1  Signifiance_Sum: 1280  Power: 0.426666666666667"
## [1] "320  Sigma: 4  n: 20  beta_1: 1.2  Signifiance_Sum: 1451  Power: 0.483666666666667"
## [1] "321  Sigma: 4  n: 20  beta_1: 1.3  Signifiance_Sum: 1660  Power: 0.553333333333333"
## [1] "322  Sigma: 4  n: 20  beta_1: 1.4  Signifiance_Sum: 1841  Power: 0.613666666666667"
## [1] "323  Sigma: 4  n: 20  beta_1: 1.5  Signifiance_Sum: 1993  Power: 0.664333333333333"
## [1] "324  Sigma: 4  n: 20  beta_1: 1.6  Signifiance_Sum: 2171  Power: 0.723666666666667"
## [1] "325  Sigma: 4  n: 20  beta_1: 1.7  Signifiance_Sum: 2349  Power: 0.783"
## [1] "326  Sigma: 4  n: 20  beta_1: 1.8  Signifiance_Sum: 2406  Power: 0.802"
## [1] "327  Sigma: 4  n: 20  beta_1: 1.9  Signifiance_Sum: 2597  Power: 0.865666666666667"
## [1] "328  Sigma: 4  n: 20  beta_1: 2  Signifiance_Sum: 2654  Power: 0.884666666666667"
## [1] "329  Sigma: 4  n: 30  beta_1: -2  Signifiance_Sum: 2923  Power: 0.974333333333333"
## [1] "330  Sigma: 4  n: 30  beta_1: -1.9  Signifiance_Sum: 2898  Power: 0.966"
## [1] "331  Sigma: 4  n: 30  beta_1: -1.8  Signifiance_Sum: 2835  Power: 0.945"
## [1] "332  Sigma: 4  n: 30  beta_1: -1.7  Signifiance_Sum: 2759  Power: 0.919666666666667"
## [1] "333  Sigma: 4  n: 30  beta_1: -1.6  Signifiance_Sum: 2632  Power: 0.877333333333333"
## [1] "334  Sigma: 4  n: 30  beta_1: -1.5  Signifiance_Sum: 2496  Power: 0.832"
## [1] "335  Sigma: 4  n: 30  beta_1: -1.4  Signifiance_Sum: 2378  Power: 0.792666666666667"
## [1] "336  Sigma: 4  n: 30  beta_1: -1.3  Signifiance_Sum: 2190  Power: 0.73"
## [1] "337  Sigma: 4  n: 30  beta_1: -1.2  Signifiance_Sum: 1966  Power: 0.655333333333333"
## [1] "338  Sigma: 4  n: 30  beta_1: -1.1  Signifiance_Sum: 1753  Power: 0.584333333333333"
## [1] "339  Sigma: 4  n: 30  beta_1: -1  Signifiance_Sum: 1534  Power: 0.511333333333333"
## [1] "340  Sigma: 4  n: 30  beta_1: -0.9  Signifiance_Sum: 1292  Power: 0.430666666666667"
## [1] "341  Sigma: 4  n: 30  beta_1: -0.8  Signifiance_Sum: 1059  Power: 0.353"
## [1] "342  Sigma: 4  n: 30  beta_1: -0.7  Signifiance_Sum: 855  Power: 0.285"
## [1] "343  Sigma: 4  n: 30  beta_1: -0.6  Signifiance_Sum: 672  Power: 0.224"
## [1] "344  Sigma: 4  n: 30  beta_1: -0.5  Signifiance_Sum: 477  Power: 0.159"
## [1] "345  Sigma: 4  n: 30  beta_1: -0.4  Signifiance_Sum: 363  Power: 0.121"
## [1] "346  Sigma: 4  n: 30  beta_1: -0.3  Signifiance_Sum: 264  Power: 0.088"
## [1] "347  Sigma: 4  n: 30  beta_1: -0.2  Signifiance_Sum: 192  Power: 0.064"
## [1] "348  Sigma: 4  n: 30  beta_1: -0.0999999999999999  Signifiance_Sum: 163  Power: 0.0543333333333333"
## [1] "349  Sigma: 4  n: 30  beta_1: 0  Signifiance_Sum: 147  Power: 0.049"
## [1] "350  Sigma: 4  n: 30  beta_1: 0.1  Signifiance_Sum: 190  Power: 0.0633333333333333"
## [1] "351  Sigma: 4  n: 30  beta_1: 0.2  Signifiance_Sum: 227  Power: 0.0756666666666667"
## [1] "352  Sigma: 4  n: 30  beta_1: 0.3  Signifiance_Sum: 243  Power: 0.081"
## [1] "353  Sigma: 4  n: 30  beta_1: 0.4  Signifiance_Sum: 389  Power: 0.129666666666667"
## [1] "354  Sigma: 4  n: 30  beta_1: 0.5  Signifiance_Sum: 514  Power: 0.171333333333333"
## [1] "355  Sigma: 4  n: 30  beta_1: 0.6  Signifiance_Sum: 655  Power: 0.218333333333333"
## [1] "356  Sigma: 4  n: 30  beta_1: 0.7  Signifiance_Sum: 883  Power: 0.294333333333333"
## [1] "357  Sigma: 4  n: 30  beta_1: 0.8  Signifiance_Sum: 1055  Power: 0.351666666666667"
## [1] "358  Sigma: 4  n: 30  beta_1: 0.9  Signifiance_Sum: 1266  Power: 0.422"
## [1] "359  Sigma: 4  n: 30  beta_1: 1  Signifiance_Sum: 1544  Power: 0.514666666666667"
## [1] "360  Sigma: 4  n: 30  beta_1: 1.1  Signifiance_Sum: 1741  Power: 0.580333333333333"
## [1] "361  Sigma: 4  n: 30  beta_1: 1.2  Signifiance_Sum: 2037  Power: 0.679"
## [1] "362  Sigma: 4  n: 30  beta_1: 1.3  Signifiance_Sum: 2200  Power: 0.733333333333333"
## [1] "363  Sigma: 4  n: 30  beta_1: 1.4  Signifiance_Sum: 2337  Power: 0.779"
## [1] "364  Sigma: 4  n: 30  beta_1: 1.5  Signifiance_Sum: 2521  Power: 0.840333333333333"
## [1] "365  Sigma: 4  n: 30  beta_1: 1.6  Signifiance_Sum: 2665  Power: 0.888333333333333"
## [1] "366  Sigma: 4  n: 30  beta_1: 1.7  Signifiance_Sum: 2755  Power: 0.918333333333333"
## [1] "367  Sigma: 4  n: 30  beta_1: 1.8  Signifiance_Sum: 2824  Power: 0.941333333333333"
## [1] "368  Sigma: 4  n: 30  beta_1: 1.9  Signifiance_Sum: 2885  Power: 0.961666666666667"
## [1] "369  Sigma: 4  n: 30  beta_1: 2  Signifiance_Sum: 2946  Power: 0.982"
par(mfrow=c(1, 2))
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="1000 Iterations",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=2,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="tan1", pch=3,lty=3)
lines(model_sim[,1], model_sim[,4], col="tan1",lty=3)

points(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)

points(model_sim[,1], model_sim[,6], col="deeppink", pch=5,lty=5)
lines(model_sim[,1], model_sim[,6], col="deeppink",lty=5)
points(model_sim[,1], model_sim[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim[,1], model_sim[,7], col="darkolivegreen4",lty=6)

points(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)

points(model_sim[,1], model_sim[,9], col="darkblue", pch=8,lty=8)
lines(model_sim[,1], model_sim[,9], col="darkblue",lty=8)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=9)

legend("topright",legend=c("σ=1,n=10","σ=1,n=20","σ=1,n=30","σ=2,n=10","σ=2,n=20","σ=2,n=30","σ=3,n=10","σ=3,n=20","σ=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))

#Below code is to plot the simulated beta_1 for 3000 simulations. This plot will helps us to understands, whether there is anything changes because of increase the simulation
#from 1000 iteration to 3000 iteration
plot(model_sim_3000[,1],model_sim_3000[,2], type="o", col="dodgerblue", pch=1, lty=1,main="3000 Iterations",xlab = "beta_1",ylab = "Power")
points(model_sim_3000[,1], model_sim_3000[,3], col="red", pch=2,lty=2)
lines(model_sim_3000[,1], model_sim_3000[,3], col="red",lty=2)
points(model_sim_3000[,1], model_sim_3000[,4], col="tan1", pch=3,lty=3)
lines(model_sim_3000[,1], model_sim_3000[,4], col="tan1",lty=3)

points(model_sim_3000[,1],model_sim_3000[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim_3000[,1],model_sim_3000[,5], type="o", col="yellow", pch=4, lty=4)

points(model_sim_3000[,1], model_sim_3000[,6], col="deeppink", pch=5,lty=5)
lines(model_sim_3000[,1], model_sim_3000[,6], col="deeppink",lty=5)
points(model_sim_3000[,1], model_sim_3000[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim_3000[,1], model_sim_3000[,7], col="darkolivegreen4",lty=6)

points(model_sim_3000[,1],model_sim_3000[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim_3000[,1],model_sim_3000[,8], type="o", col="darkgreen", pch=7, lty=7)

points(model_sim_3000[,1], model_sim_3000[,9], col="darkblue", pch=8,lty=8)
lines(model_sim_3000[,1], model_sim_3000[,9], col="darkblue",lty=8)
points(model_sim_3000[,1], model_sim_3000[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim_3000[,1], model_sim_3000[,10], col="blueviolet",lty=9)

legend("topright",legend=c("σ=1,n=10","σ=1,n=20","σ=1,n=30","σ=2,n=10","σ=2,n=20","σ=2,n=30","σ=3,n=10","σ=3,n=20","σ=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))

From the above plots its clearly visible that the plots for the 3000 iterations have much smoother curve with respect to the Power value, as compared to the 1000 iterations (where the power value is ups-down). Let’s take the range of the Power for the 1000 vs 3000 iterations for a given value of sigma (let’s say , sigma = 1) and for a particular value of sample size ( n= 10).

  • Power for 1000 Iteration (sigma = 1, n = 10) is between 0.041 and 1
  • Power for 3000 Iteration (sigma = 1, n = 10)is between 0.0533333 and 1

From the range prospective the 3000 iterative have slightly increased value as compared to 1000 iteration. Hence if we increase the iteration, the Power becomes more and more accurate to the Truth, hence I conclude that 1000 iteration is sufficient to know the impact of beta(β1), sample size(n) and sigma(σ) on the Power, however 1000 iteration is not sufficient to know the truth of Power